Algebra Examples

$\frac{{(x - h)}^{2}}{a^{2}} - \frac{{(y - k)}^{2}}{b^{2}} = 1$

Step 1

Add $\frac{{(y - k)}^{2}}{b^{2}}$ to both sides of the equation.

$\frac{{(x - h)}^{2}}{a^{2}} = 1 + \frac{{(y - k)}^{2}}{b^{2}}$

Step 2

Move all terms containing variables to the left side of the equation.

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Step 2.1

Subtract $\frac{{(y - k)}^{2}}{b^{2}}$ from both sides of the equation.

$\frac{{(x - h)}^{2}}{a^{2}} - \frac{{(y - k)}^{2}}{b^{2}} = 1$

Step 2.2

To write $\frac{{(x - h)}^{2}}{a^{2}}$ as a fraction with a common denominator, multiply by $\frac{b^{2}}{b^{2}}$ .

$\frac{{(x - h)}^{2}}{a^{2}} \cdot \frac{b^{2}}{b^{2}} - \frac{{(y - k)}^{2}}{b^{2}} = 1$

Step 2.3

To write $- \frac{{(y - k)}^{2}}{b^{2}}$ as a fraction with a common denominator, multiply by $\frac{a^{2}}{a^{2}}$ .

$\frac{{(x - h)}^{2}}{a^{2}} \cdot \frac{b^{2}}{b^{2}} - \frac{{(y - k)}^{2}}{b^{2}} \cdot \frac{a^{2}}{a^{2}} = 1$

Step 2.4

Write each expression with a common denominator of $a^{2} b^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.4.1

Multiply $\frac{{(x - h)}^{2}}{a^{2}}$ by $\frac{b^{2}}{b^{2}}$ .

$\frac{{(x - h)}^{2} b^{2}}{a^{2} b^{2}} - \frac{{(y - k)}^{2}}{b^{2}} \cdot \frac{a^{2}}{a^{2}} = 1$

Step 2.4.2

Multiply $\frac{{(y - k)}^{2}}{b^{2}}$ by $\frac{a^{2}}{a^{2}}$ .

$\frac{{(x - h)}^{2} b^{2}}{a^{2} b^{2}} - \frac{{(y - k)}^{2} a^{2}}{b^{2} a^{2}} = 1$

Step 2.4.3

Reorder the factors of $b^{2} a^{2}$ .

$\frac{{(x - h)}^{2} b^{2}}{a^{2} b^{2}} - \frac{{(y - k)}^{2} a^{2}}{a^{2} b^{2}} = 1$

Step 2.5

Combine the numerators over the common denominator.

$\frac{{(x - h)}^{2} b^{2} - {(y - k)}^{2} a^{2}}{a^{2} b^{2}} = 1$

Step 2.6

Simplify the numerator.

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Step 2.6.1

Rewrite ${(x - h)}^{2} b^{2}$ as ${((x - h) b)}^{2}$ .

$\frac{{((x - h) b)}^{2} - {(y - k)}^{2} a^{2}}{a^{2} b^{2}} = 1$

Step 2.6.2

Rewrite ${(y - k)}^{2} a^{2}$ as ${((y - k) a)}^{2}$ .

$\frac{{((x - h) b)}^{2} - {((y - k) a)}^{2}}{a^{2} b^{2}} = 1$

Step 2.6.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = (x - h) b$ and $b = (y - k) a$ .

$\frac{((x - h) b + (y - k) a) ((x - h) b - ((y - k) a))}{a^{2} b^{2}} = 1$

Step 2.6.4

Simplify.

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Step 2.6.4.1

Apply the distributive property.

$\frac{(x b - h b + (y - k) a) ((x - h) b - ((y - k) a))}{a^{2} b^{2}} = 1$

Step 2.6.4.2

Apply the distributive property.

$\frac{(x b - h b + y a - k a) ((x - h) b - ((y - k) a))}{a^{2} b^{2}} = 1$

Step 2.6.4.3

Apply the distributive property.

$\frac{(x b - h b + y a - k a) (x b - h b - ((y - k) a))}{a^{2} b^{2}} = 1$

Step 2.6.4.4

Apply the distributive property.

$\frac{(x b - h b + y a - k a) (x b - h b - (y a - k a))}{a^{2} b^{2}} = 1$

Step 2.6.4.5

Apply the distributive property.

$\frac{(x b - h b + y a - k a) (x b - h b - (y a) - (- k a))}{a^{2} b^{2}} = 1$

Step 2.6.4.6

Multiply $- (- k a)$ .

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Step 2.6.4.6.1

Multiply $- 1$ by $- 1$ .

$\frac{(x b - h b + y a - k a) (x b - h b - y a + 1 (k a))}{a^{2} b^{2}} = 1$

Step 2.6.4.6.2

Multiply $k$ by $1$ .

$\frac{(x b - h b + y a - k a) (x b - h b - y a + k a)}{a^{2} b^{2}} = 1$

Step 3

Multiply both sides by $a^{2} b^{2}$ .

$\frac{(x b - h b + y a - k a) (x b - h b - y a + k a)}{a^{2} b^{2}} (a^{2} b^{2}) = 1 (a^{2} b^{2})$

Step 4

Simplify.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Simplify $\frac{(x b - h b + y a - k a) (x b - h b - y a + k a)}{a^{2} b^{2}} (a^{2} b^{2})$ .

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Step 4.1.1.1

Cancel the common factor of $a^{2} b^{2}$ .

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Step 4.1.1.1.1

Cancel the common factor.

$\frac{(x b - h b + y a - k a) (x b - h b - y a + k a)}{a^{2} b^{2}} (a^{2} b^{2}) = 1 (a^{2} b^{2})$

Step 4.1.1.1.2

Rewrite the expression.

$(x b - h b + y a - k a) (x b - h b - y a + k a) = 1 (a^{2} b^{2})$

Step 4.1.1.2

Expand $(x b - h b + y a - k a) (x b - h b - y a + k a)$ by multiplying each term in the first expression by each term in the second expression.

$x b (x b) + x b (- h b) + x b (- y a) + x b (k a) - h b (x b) - h b (- h b) - h b (- y a) - h b (k a) + y a (x b) + y a (- h b) + y a (- y a) + y a (k a) - k a (x b) - k a (- h b) - k a (- y a) - k a (k a) = 1 (a^{2} b^{2})$

Step 4.1.1.3

Simplify terms.

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Step 4.1.1.3.1

Combine the opposite terms in $x b (x b) + x b (- h b) + x b (- y a) + x b (k a) - h b (x b) - h b (- h b) - h b (- y a) - h b (k a) + y a (x b) + y a (- h b) + y a (- y a) + y a (k a) - k a (x b) - k a (- h b) - k a (- y a) - k a (k a)$ .

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Step 4.1.1.3.1.1

Reorder the factors in the terms $x b (- y a)$ and $y a (x b)$ .

$x b (x b) + x b (- h b) - a b x y + x b (k a) - h b (x b) - h b (- h b) - h b (- y a) - h b (k a) + a b x y + y a (- h b) + y a (- y a) + y a (k a) - k a (x b) - k a (- h b) - k a (- y a) - k a (k a) = 1 (a^{2} b^{2})$

Step 4.1.1.3.1.2

Add $- a b x y$ and $a b x y$ .

$x b (x b) + x b (- h b) + x b (k a) - h b (x b) - h b (- h b) - h b (- y a) - h b (k a) + 0 + y a (- h b) + y a (- y a) + y a (k a) - k a (x b) - k a (- h b) - k a (- y a) - k a (k a) = 1 (a^{2} b^{2})$

Step 4.1.1.3.1.3

Add $x b (x b) + x b (- h b) + x b (k a) - h b (x b) - h b (- h b) - h b (- y a) - h b (k a)$ and $0$ .

$x b (x b) + x b (- h b) + x b (k a) - h b (x b) - h b (- h b) - h b (- y a) - h b (k a) + y a (- h b) + y a (- y a) + y a (k a) - k a (x b) - k a (- h b) - k a (- y a) - k a (k a) = 1 (a^{2} b^{2})$

Step 4.1.1.3.1.4

Reorder the factors in the terms $x b (k a)$ and $- k a (x b)$ .

$x b (x b) + x b (- h b) + a b k x - h b (x b) - h b (- h b) - h b (- y a) - h b (k a) + y a (- h b) + y a (- y a) + y a (k a) - a b k x - k a (- h b) - k a (- y a) - k a (k a) = 1 (a^{2} b^{2})$

Step 4.1.1.3.1.5

Subtract $a b k x$ from $a b k x$ .

$x b (x b) + x b (- h b) - h b (x b) - h b (- h b) - h b (- y a) - h b (k a) + y a (- h b) + y a (- y a) + y a (k a) + 0 - k a (- h b) - k a (- y a) - k a (k a) = 1 (a^{2} b^{2})$

Step 4.1.1.3.1.6

Add $x b (x b) + x b (- h b) - h b (x b) - h b (- h b) - h b (- y a) - h b (k a) + y a (- h b) + y a (- y a) + y a (k a)$ and $0$ .

$x b (x b) + x b (- h b) - h b (x b) - h b (- h b) - h b (- y a) - h b (k a) + y a (- h b) + y a (- y a) + y a (k a) - k a (- h b) - k a (- y a) - k a (k a) = 1 (a^{2} b^{2})$

Step 4.1.1.3.2

Simplify each term.

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Step 4.1.1.3.2.1

Multiply $x$ by $x$ by adding the exponents.

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Step 4.1.1.3.2.1.1

Move $x$ .

$x \cdot x b \cdot b + x b (- h b) - h b (x b) - h b (- h b) - h b (- y a) - h b (k a) + y a (- h b) + y a (- y a) + y a (k a) - k a (- h b) - k a (- y a) - k a (k a) = 1 (a^{2} b^{2})$

Step 4.1.1.3.2.1.2

Multiply $x$ by $x$ .

$x^{2} b \cdot b + x b (- h b) - h b (x b) - h b (- h b) - h b (- y a) - h b (k a) + y a (- h b) + y a (- y a) + y a (k a) - k a (- h b) - k a (- y a) - k a (k a) = 1 (a^{2} b^{2})$

Step 4.1.1.3.2.2

Multiply $b$ by $b$ by adding the exponents.

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Step 4.1.1.3.2.2.1

Move $b$ .

$x^{2} (b \cdot b) + x b (- h b) - h b (x b) - h b (- h b) - h b (- y a) - h b (k a) + y a (- h b) + y a (- y a) + y a (k a) - k a (- h b) - k a (- y a) - k a (k a) = 1 (a^{2} b^{2})$

Step 4.1.1.3.2.2.2

Multiply $b$ by $b$ .

$x^{2} b^{2} + x b (- h b) - h b (x b) - h b (- h b) - h b (- y a) - h b (k a) + y a (- h b) + y a (- y a) + y a (k a) - k a (- h b) - k a (- y a) - k a (k a) = 1 (a^{2} b^{2})$

Step 4.1.1.3.2.3

Rewrite using the commutative property of multiplication.

$x^{2} b^{2} - x b (h b) - h b (x b) - h b (- h b) - h b (- y a) - h b (k a) + y a (- h b) + y a (- y a) + y a (k a) - k a (- h b) - k a (- y a) - k a (k a) = 1 (a^{2} b^{2})$

Step 4.1.1.3.2.4

Multiply $b$ by $b$ by adding the exponents.

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Step 4.1.1.3.2.4.1

Move $b$ .

$x^{2} b^{2} - x (b \cdot b) h - h b (x b) - h b (- h b) - h b (- y a) - h b (k a) + y a (- h b) + y a (- y a) + y a (k a) - k a (- h b) - k a (- y a) - k a (k a) = 1 (a^{2} b^{2})$

Step 4.1.1.3.2.4.2

Multiply $b$ by $b$ .

$x^{2} b^{2} - x b^{2} h - h b (x b) - h b (- h b) - h b (- y a) - h b (k a) + y a (- h b) + y a (- y a) + y a (k a) - k a (- h b) - k a (- y a) - k a (k a) = 1 (a^{2} b^{2})$

Step 4.1.1.3.2.5

Multiply $b$ by $b$ by adding the exponents.

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Step 4.1.1.3.2.5.1

Move $b$ .

$x^{2} b^{2} - x b^{2} h - h (b \cdot b) x - h b (- h b) - h b (- y a) - h b (k a) + y a (- h b) + y a (- y a) + y a (k a) - k a (- h b) - k a (- y a) - k a (k a) = 1 (a^{2} b^{2})$

Step 4.1.1.3.2.5.2

Multiply $b$ by $b$ .

$x^{2} b^{2} - x b^{2} h - h b^{2} x - h b (- h b) - h b (- y a) - h b (k a) + y a (- h b) + y a (- y a) + y a (k a) - k a (- h b) - k a (- y a) - k a (k a) = 1 (a^{2} b^{2})$

Step 4.1.1.3.2.6

Multiply $h$ by $h$ by adding the exponents.

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Step 4.1.1.3.2.6.1

Move $h$ .

$x^{2} b^{2} - x b^{2} h - h b^{2} x - (h \cdot h) b (- 1 b) - h b (- y a) - h b (k a) + y a (- h b) + y a (- y a) + y a (k a) - k a (- h b) - k a (- y a) - k a (k a) = 1 (a^{2} b^{2})$

Step 4.1.1.3.2.6.2

Multiply $h$ by $h$ .

$x^{2} b^{2} - x b^{2} h - h b^{2} x - h^{2} b (- 1 b) - h b (- y a) - h b (k a) + y a (- h b) + y a (- y a) + y a (k a) - k a (- h b) - k a (- y a) - k a (k a) = 1 (a^{2} b^{2})$

Step 4.1.1.3.2.7

Multiply $b$ by $b$ by adding the exponents.

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Step 4.1.1.3.2.7.1

Move $b$ .

$x^{2} b^{2} - x b^{2} h - h b^{2} x - h^{2} (b \cdot b) \cdot - 1 - h b (- y a) - h b (k a) + y a (- h b) + y a (- y a) + y a (k a) - k a (- h b) - k a (- y a) - k a (k a) = 1 (a^{2} b^{2})$

Step 4.1.1.3.2.7.2

Multiply $b$ by $b$ .

$x^{2} b^{2} - x b^{2} h - h b^{2} x - h^{2} b^{2} \cdot - 1 - h b (- y a) - h b (k a) + y a (- h b) + y a (- y a) + y a (k a) - k a (- h b) - k a (- y a) - k a (k a) = 1 (a^{2} b^{2})$

Step 4.1.1.3.2.8

Multiply $- h^{2} b^{2} \cdot - 1$ .

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Step 4.1.1.3.2.8.1

Multiply $- 1$ by $- 1$ .

$x^{2} b^{2} - x b^{2} h - h b^{2} x + 1 h^{2} b^{2} - h b (- y a) - h b (k a) + y a (- h b) + y a (- y a) + y a (k a) - k a (- h b) - k a (- y a) - k a (k a) = 1 (a^{2} b^{2})$

Step 4.1.1.3.2.8.2

Multiply $h^{2}$ by $1$ .

$x^{2} b^{2} - x b^{2} h - h b^{2} x + h^{2} b^{2} - h b (- y a) - h b (k a) + y a (- h b) + y a (- y a) + y a (k a) - k a (- h b) - k a (- y a) - k a (k a) = 1 (a^{2} b^{2})$

Step 4.1.1.3.2.9

Multiply $- h b (- y a)$ .

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Step 4.1.1.3.2.9.1

Multiply $- 1$ by $- 1$ .

$x^{2} b^{2} - x b^{2} h - h b^{2} x + h^{2} b^{2} + 1 h b (y a) - h b (k a) + y a (- h b) + y a (- y a) + y a (k a) - k a (- h b) - k a (- y a) - k a (k a) = 1 (a^{2} b^{2})$

Step 4.1.1.3.2.9.2

Multiply $h$ by $1$ .

$x^{2} b^{2} - x b^{2} h - h b^{2} x + h^{2} b^{2} + h b (y a) - h b (k a) + y a (- h b) + y a (- y a) + y a (k a) - k a (- h b) - k a (- y a) - k a (k a) = 1 (a^{2} b^{2})$

Step 4.1.1.3.2.10

Rewrite using the commutative property of multiplication.

$x^{2} b^{2} - x b^{2} h - h b^{2} x + h^{2} b^{2} + h b y a - h b k a - y a h b + y a (- y a) + y a (k a) - k a (- h b) - k a (- y a) - k a (k a) = 1 (a^{2} b^{2})$

Step 4.1.1.3.2.11

Rewrite using the commutative property of multiplication.

$x^{2} b^{2} - x b^{2} h - h b^{2} x + h^{2} b^{2} + h b y a - h b k a - y a h b - y a (y a) + y a (k a) - k a (- h b) - k a (- y a) - k a (k a) = 1 (a^{2} b^{2})$

Step 4.1.1.3.2.12

Multiply $y$ by $y$ by adding the exponents.

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Step 4.1.1.3.2.12.1

Move $y$ .

$x^{2} b^{2} - x b^{2} h - h b^{2} x + h^{2} b^{2} + h b y a - h b k a - y a h b - (y \cdot y) a \cdot a + y a (k a) - k a (- h b) - k a (- y a) - k a (k a) = 1 (a^{2} b^{2})$

Step 4.1.1.3.2.12.2

Multiply $y$ by $y$ .

$x^{2} b^{2} - x b^{2} h - h b^{2} x + h^{2} b^{2} + h b y a - h b k a - y a h b - y^{2} a \cdot a + y a (k a) - k a (- h b) - k a (- y a) - k a (k a) = 1 (a^{2} b^{2})$

Step 4.1.1.3.2.13

Multiply $a$ by $a$ by adding the exponents.

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Step 4.1.1.3.2.13.1

Move $a$ .

$x^{2} b^{2} - x b^{2} h - h b^{2} x + h^{2} b^{2} + h b y a - h b k a - y a h b - y^{2} (a \cdot a) + y a (k a) - k a (- h b) - k a (- y a) - k a (k a) = 1 (a^{2} b^{2})$

Step 4.1.1.3.2.13.2

Multiply $a$ by $a$ .

$x^{2} b^{2} - x b^{2} h - h b^{2} x + h^{2} b^{2} + h b y a - h b k a - y a h b - y^{2} a^{2} + y a (k a) - k a (- h b) - k a (- y a) - k a (k a) = 1 (a^{2} b^{2})$

Step 4.1.1.3.2.14

Multiply $a$ by $a$ by adding the exponents.

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Step 4.1.1.3.2.14.1

Move $a$ .

$x^{2} b^{2} - x b^{2} h - h b^{2} x + h^{2} b^{2} + h b y a - h b k a - y a h b - y^{2} a^{2} + y (a \cdot a) k - k a (- h b) - k a (- y a) - k a (k a) = 1 (a^{2} b^{2})$

Step 4.1.1.3.2.14.2

Multiply $a$ by $a$ .

$x^{2} b^{2} - x b^{2} h - h b^{2} x + h^{2} b^{2} + h b y a - h b k a - y a h b - y^{2} a^{2} + y a^{2} k - k a (- h b) - k a (- y a) - k a (k a) = 1 (a^{2} b^{2})$

Step 4.1.1.3.2.15

Multiply $- k a (- h b)$ .

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Step 4.1.1.3.2.15.1

Multiply $- 1$ by $- 1$ .

$x^{2} b^{2} - x b^{2} h - h b^{2} x + h^{2} b^{2} + h b y a - h b k a - y a h b - y^{2} a^{2} + y a^{2} k + 1 k a (h b) - k a (- y a) - k a (k a) = 1 (a^{2} b^{2})$

Step 4.1.1.3.2.15.2

Multiply $k$ by $1$ .

$x^{2} b^{2} - x b^{2} h - h b^{2} x + h^{2} b^{2} + h b y a - h b k a - y a h b - y^{2} a^{2} + y a^{2} k + k a (h b) - k a (- y a) - k a (k a) = 1 (a^{2} b^{2})$

Step 4.1.1.3.2.16

Multiply $a$ by $a$ by adding the exponents.

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Step 4.1.1.3.2.16.1

Move $a$ .

$x^{2} b^{2} - x b^{2} h - h b^{2} x + h^{2} b^{2} + h b y a - h b k a - y a h b - y^{2} a^{2} + y a^{2} k + k a h b - k (a \cdot a) (- y) - k a (k a) = 1 (a^{2} b^{2})$

Step 4.1.1.3.2.16.2

Multiply $a$ by $a$ .

$x^{2} b^{2} - x b^{2} h - h b^{2} x + h^{2} b^{2} + h b y a - h b k a - y a h b - y^{2} a^{2} + y a^{2} k + k a h b - k a^{2} (- y) - k a (k a) = 1 (a^{2} b^{2})$

Step 4.1.1.3.2.17

Multiply $- k a^{2} (- y)$ .

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Step 4.1.1.3.2.17.1

Multiply $- 1$ by $- 1$ .

$x^{2} b^{2} - x b^{2} h - h b^{2} x + h^{2} b^{2} + h b y a - h b k a - y a h b - y^{2} a^{2} + y a^{2} k + k a h b + 1 k a^{2} y - k a (k a) = 1 (a^{2} b^{2})$

Step 4.1.1.3.2.17.2

Multiply $k$ by $1$ .

$x^{2} b^{2} - x b^{2} h - h b^{2} x + h^{2} b^{2} + h b y a - h b k a - y a h b - y^{2} a^{2} + y a^{2} k + k a h b + k a^{2} y - k a (k a) = 1 (a^{2} b^{2})$

Step 4.1.1.3.2.18

Multiply $k$ by $k$ by adding the exponents.

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Step 4.1.1.3.2.18.1

Move $k$ .

$x^{2} b^{2} - x b^{2} h - h b^{2} x + h^{2} b^{2} + h b y a - h b k a - y a h b - y^{2} a^{2} + y a^{2} k + k a h b + k a^{2} y - (k \cdot k) a \cdot a = 1 (a^{2} b^{2})$

Step 4.1.1.3.2.18.2

Multiply $k$ by $k$ .

$x^{2} b^{2} - x b^{2} h - h b^{2} x + h^{2} b^{2} + h b y a - h b k a - y a h b - y^{2} a^{2} + y a^{2} k + k a h b + k a^{2} y - k^{2} a \cdot a = 1 (a^{2} b^{2})$

Step 4.1.1.3.2.19

Multiply $a$ by $a$ by adding the exponents.

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Step 4.1.1.3.2.19.1

Move $a$ .

$x^{2} b^{2} - x b^{2} h - h b^{2} x + h^{2} b^{2} + h b y a - h b k a - y a h b - y^{2} a^{2} + y a^{2} k + k a h b + k a^{2} y - k^{2} (a \cdot a) = 1 (a^{2} b^{2})$

Step 4.1.1.3.2.19.2

Multiply $a$ by $a$ .

$x^{2} b^{2} - x b^{2} h - h b^{2} x + h^{2} b^{2} + h b y a - h b k a - y a h b - y^{2} a^{2} + y a^{2} k + k a h b + k a^{2} y - k^{2} a^{2} = 1 (a^{2} b^{2})$

Step 4.1.1.3.3

Simplify by adding terms.

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Step 4.1.1.3.3.1

Combine the opposite terms in $x^{2} b^{2} - x b^{2} h - h b^{2} x + h^{2} b^{2} + h b y a - h b k a - y a h b - y^{2} a^{2} + y a^{2} k + k a h b + k a^{2} y - k^{2} a^{2}$ .

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Step 4.1.1.3.3.1.1

Reorder the factors in the terms $h b y a$ and $- y a h b$ .

$x^{2} b^{2} - x b^{2} h - h b^{2} x + h^{2} b^{2} + a b h y - h b k a - a b h y - y^{2} a^{2} + y a^{2} k + k a h b + k a^{2} y - k^{2} a^{2} = 1 (a^{2} b^{2})$

Step 4.1.1.3.3.1.2

Subtract $a b h y$ from $a b h y$ .

$x^{2} b^{2} - x b^{2} h - h b^{2} x + h^{2} b^{2} - h b k a + 0 - y^{2} a^{2} + y a^{2} k + k a h b + k a^{2} y - k^{2} a^{2} = 1 (a^{2} b^{2})$

Step 4.1.1.3.3.1.3

Add $x^{2} b^{2} - x b^{2} h - h b^{2} x + h^{2} b^{2} - h b k a$ and $0$ .

$x^{2} b^{2} - x b^{2} h - h b^{2} x + h^{2} b^{2} - h b k a - y^{2} a^{2} + y a^{2} k + k a h b + k a^{2} y - k^{2} a^{2} = 1 (a^{2} b^{2})$

Step 4.1.1.3.3.1.4

Reorder the factors in the terms $- h b k a$ and $k a h b$ .

$x^{2} b^{2} - x b^{2} h - h b^{2} x + h^{2} b^{2} - a b h k - y^{2} a^{2} + y a^{2} k + a b h k + k a^{2} y - k^{2} a^{2} = 1 (a^{2} b^{2})$

Step 4.1.1.3.3.1.5

Add $- a b h k$ and $a b h k$ .

$x^{2} b^{2} - x b^{2} h - h b^{2} x + h^{2} b^{2} - y^{2} a^{2} + y a^{2} k + 0 + k a^{2} y - k^{2} a^{2} = 1 (a^{2} b^{2})$

Step 4.1.1.3.3.1.6

Add $x^{2} b^{2} - x b^{2} h - h b^{2} x + h^{2} b^{2} - y^{2} a^{2} + y a^{2} k$ and $0$ .

$x^{2} b^{2} - x b^{2} h - h b^{2} x + h^{2} b^{2} - y^{2} a^{2} + y a^{2} k + k a^{2} y - k^{2} a^{2} = 1 (a^{2} b^{2})$

Step 4.1.1.3.3.2

Simplify the expression.

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Step 4.1.1.3.3.2.1

Move $h$ .

$x^{2} b^{2} - b^{2} h x - 1 b^{2} h x + h^{2} b^{2} - y^{2} a^{2} + y a^{2} k + k a^{2} y - k^{2} a^{2} = 1 (a^{2} b^{2})$

Step 4.1.1.3.3.2.2

Rewrite $- 1 b^{2}$ as $- b^{2}$ .

$x^{2} b^{2} - b^{2} h x - b^{2} h x + h^{2} b^{2} - y^{2} a^{2} + y a^{2} k + k a^{2} y - k^{2} a^{2} = 1 (a^{2} b^{2})$

Step 4.1.1.3.3.3

Subtract $b^{2} h x$ from $- b^{2} h x$ .

$x^{2} b^{2} - 2 b^{2} h x + h^{2} b^{2} - y^{2} a^{2} + y a^{2} k + k a^{2} y - k^{2} a^{2} = 1 (a^{2} b^{2})$

Step 4.1.1.3.3.4

Reorder $k$ and $a^{2}$ .

$x^{2} b^{2} - 2 b^{2} h x + h^{2} b^{2} - y^{2} a^{2} + a^{2} k y + a^{2} k y - k^{2} a^{2} = 1 (a^{2} b^{2})$

Step 4.1.1.3.3.5

Add $a^{2} k y$ and $a^{2} k y$ .

$x^{2} b^{2} - 2 b^{2} h x + h^{2} b^{2} - y^{2} a^{2} + 2 a^{2} k y - k^{2} a^{2} = 1 (a^{2} b^{2})$

Step 4.1.1.3.3.6

Simplify the expression.

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Step 4.1.1.3.3.6.1

Reorder $x^{2}$ and $b^{2}$ .

$b^{2} x^{2} - 2 b^{2} h x + h^{2} b^{2} - y^{2} a^{2} + 2 a^{2} k y - k^{2} a^{2} = 1 (a^{2} b^{2})$

Step 4.1.1.3.3.6.2

Reorder $h^{2}$ and $b^{2}$ .

$b^{2} x^{2} - 2 b^{2} h x + b^{2} h^{2} - y^{2} a^{2} + 2 a^{2} k y - k^{2} a^{2} = 1 (a^{2} b^{2})$

Step 4.1.1.3.3.6.3

Move $y^{2}$ .

$b^{2} x^{2} - 2 b^{2} h x + b^{2} h^{2} - a^{2} y^{2} + 2 a^{2} k y - k^{2} a^{2} = 1 (a^{2} b^{2})$

Step 4.1.1.3.3.6.4

Move $k^{2}$ .

$b^{2} x^{2} - 2 b^{2} h x + b^{2} h^{2} - a^{2} y^{2} + 2 a^{2} k y - 1 a^{2} k^{2} = 1 (a^{2} b^{2})$

Step 4.1.1.3.3.6.5

Move $b^{2} x^{2}$ .

$- 2 b^{2} h x + b^{2} h^{2} - a^{2} y^{2} + 2 a^{2} k y - 1 a^{2} k^{2} + b^{2} x^{2} = 1 (a^{2} b^{2})$

Step 4.1.1.3.3.6.6

Move $- 2 b^{2} h x$ .

$b^{2} h^{2} - a^{2} y^{2} + 2 a^{2} k y - 1 a^{2} k^{2} - 2 b^{2} h x + b^{2} x^{2} = 1 (a^{2} b^{2})$

Step 4.1.1.3.3.6.7

Move $b^{2} h^{2}$ .

$- a^{2} y^{2} + 2 a^{2} k y - 1 a^{2} k^{2} + b^{2} h^{2} - 2 b^{2} h x + b^{2} x^{2} = 1 (a^{2} b^{2})$

Step 4.1.1.3.3.6.8

Move $- a^{2} y^{2}$ .

$2 a^{2} k y - 1 a^{2} k^{2} - a^{2} y^{2} + b^{2} h^{2} - 2 b^{2} h x + b^{2} x^{2} = 1 (a^{2} b^{2})$

Step 4.1.1.3.3.6.9

Reorder $2 a^{2} k y$ and $- 1 a^{2} k^{2}$ .

$- 1 a^{2} k^{2} + 2 a^{2} k y - a^{2} y^{2} + b^{2} h^{2} - 2 b^{2} h x + b^{2} x^{2} = 1 (a^{2} b^{2})$

Step 4.2

Simplify the right side.

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Step 4.2.1

Multiply $a^{2} b^{2}$ by $1$ .

$- 1 a^{2} k^{2} + 2 a^{2} k y - a^{2} y^{2} + b^{2} h^{2} - 2 b^{2} h x + b^{2} x^{2} = a^{2} b^{2}$

Step 5

Solve for $x$ .

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Step 5.1

Rewrite $- 1 a^{2}$ as $- a^{2}$ .

$- a^{2} k^{2} + 2 a^{2} k y - a^{2} y^{2} + b^{2} h^{2} - 2 b^{2} h x + b^{2} x^{2} = a^{2} b^{2}$

Step 5.2

Subtract $a^{2} b^{2}$ from both sides of the equation.

$- a^{2} k^{2} + 2 a^{2} k y - a^{2} y^{2} + b^{2} h^{2} - 2 b^{2} h x + b^{2} x^{2} - a^{2} b^{2} = 0$

Step 5.3

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 5.4

Substitute the values $a = b^{2}$ , $b = - 2 b^{2} h$ , and $c = - a^{2} k^{2} + 2 a^{2} k y - a^{2} y^{2} + b^{2} h^{2} - a^{2} b^{2}$ into the quadratic formula and solve for $x$ .

$\frac{2 b^{2} h \pm \sqrt{{(- 2 b^{2} h)}^{2} - 4 \cdot (b^{2} \cdot (- a^{2} k^{2} + 2 a^{2} k y - a^{2} y^{2} + b^{2} h^{2} - a^{2} b^{2}))}}{2 b^{2}}$

Step 5.5

Simplify.

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Step 5.5.1

Simplify the numerator.

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Step 5.5.1.1

Rewrite $a^{2} b^{2}$ as ${(a b)}^{2}$ .

$x = \frac{2 b^{2} h \pm \sqrt{{(- 2 (b^{2} h))}^{2} - 4 \cdot (b^{2} \cdot (- {(a k)}^{2} + 2 (a^{2} (k y)) - {(a y)}^{2} + {(b h)}^{2} - {(a b)}^{2}))}}{2 b^{2}}$

Step 5.5.1.2

Let $u = b^{2} \cdot (- {(a k)}^{2} + 2 (a^{2} (k y)) - {(a y)}^{2} + {(b h)}^{2} - {(a b)}^{2})$ . Substitute $u$ for all occurrences of $b^{2} \cdot (- {(a k)}^{2} + 2 (a^{2} (k y)) - {(a y)}^{2} + {(b h)}^{2} - {(a b)}^{2})$ .

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Step 5.5.1.2.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 5.5.1.2.1.1

Apply the product rule to $- 2 b^{2} h$ .

$x = \frac{2 b^{2} h \pm \sqrt{{(- 2 b^{2})}^{2} h^{2} - 4 \cdot u}}{2 b^{2}}$

Step 5.5.1.2.1.2

Apply the product rule to $- 2 b^{2}$ .

$x = \frac{2 b^{2} h \pm \sqrt{{(- 2)}^{2} {(b^{2})}^{2} h^{2} - 4 \cdot u}}{2 b^{2}}$

Step 5.5.1.2.2

Raise $- 2$ to the power of $2$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 {(b^{2})}^{2} h^{2} - 4 \cdot u}}{2 b^{2}}$

Step 5.5.1.2.3

Multiply the exponents in ${(b^{2})}^{2}$ .

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Step 5.5.1.2.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 b^{2 \cdot 2} h^{2} - 4 \cdot u}}{2 b^{2}}$

Step 5.5.1.2.3.2

Multiply $2$ by $2$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 b^{4} h^{2} - 4 \cdot u}}{2 b^{2}}$

$x = \frac{2 b^{2} h \pm \sqrt{4 b^{4} h^{2} - 4 u}}{2 b^{2}}$

Step 5.5.1.3

Factor $4$ out of $4 b^{4} h^{2} - 4 u$ .

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Step 5.5.1.3.1

Factor $4$ out of $4 b^{4} h^{2}$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{4} h^{2}) - 4 u}}{2 b^{2}}$

Step 5.5.1.3.2

Factor $4$ out of $- 4 u$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{4} h^{2}) + 4 (- u)}}{2 b^{2}}$

Step 5.5.1.3.3

Factor $4$ out of $4 (b^{4} h^{2}) + 4 (- u)$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{4} h^{2} - u)}}{2 b^{2}}$

Step 5.5.1.4

Replace all occurrences of $u$ with $b^{2} \cdot (- {(a k)}^{2} + 2 (a^{2} (k y)) - {(a y)}^{2} + {(b h)}^{2} - {(a b)}^{2})$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{4} h^{2} - (b^{2} \cdot (- {(a k)}^{2} + 2 (a^{2} (k y)) - {(a y)}^{2} + {(b h)}^{2} - {(a b)}^{2})))}}{2 b^{2}}$

Step 5.5.1.5

Simplify.

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Step 5.5.1.5.1

Simplify each term.

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Step 5.5.1.5.1.1

Simplify each term.

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Step 5.5.1.5.1.1.1

Apply the product rule to $a k$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{4} h^{2} - (b^{2} \cdot (- (a^{2} k^{2}) + 2 (a^{2} (k y)) - {(a y)}^{2} + {(b h)}^{2} - {(a b)}^{2})))}}{2 b^{2}}$

Step 5.5.1.5.1.1.2

Apply the product rule to $a y$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{4} h^{2} - (b^{2} \cdot (- a^{2} k^{2} + 2 a^{2} k y - (a^{2} y^{2}) + {(b h)}^{2} - {(a b)}^{2})))}}{2 b^{2}}$

Step 5.5.1.5.1.1.3

Apply the product rule to $b h$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{4} h^{2} - (b^{2} \cdot (- a^{2} k^{2} + 2 a^{2} k y - a^{2} y^{2} + b^{2} h^{2} - {(a b)}^{2})))}}{2 b^{2}}$

Step 5.5.1.5.1.1.4

Apply the product rule to $a b$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{4} h^{2} - (b^{2} \cdot (- a^{2} k^{2} + 2 a^{2} k y - a^{2} y^{2} + b^{2} h^{2} - a^{2} b^{2})))}}{2 b^{2}}$

Step 5.5.1.5.1.2

Apply the distributive property.

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{4} h^{2} - (b^{2} (- a^{2} k^{2}) + b^{2} (2 a^{2} k y) + b^{2} (- a^{2} y^{2}) + b^{2} (b^{2} h^{2}) + b^{2} (- a^{2} b^{2})))}}{2 b^{2}}$

Step 5.5.1.5.1.3

Simplify.

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Step 5.5.1.5.1.3.1

Rewrite using the commutative property of multiplication.

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{4} h^{2} - (- b^{2} a^{2} k^{2} + b^{2} (2 a^{2} k y) + b^{2} (- a^{2} y^{2}) + b^{2} (b^{2} h^{2}) + b^{2} (- a^{2} b^{2})))}}{2 b^{2}}$

Step 5.5.1.5.1.3.2

Rewrite using the commutative property of multiplication.

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{4} h^{2} - (- b^{2} a^{2} k^{2} + 2 b^{2} a^{2} k y + b^{2} (- a^{2} y^{2}) + b^{2} (b^{2} h^{2}) + b^{2} (- a^{2} b^{2})))}}{2 b^{2}}$

Step 5.5.1.5.1.3.3

Rewrite using the commutative property of multiplication.

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{4} h^{2} - (- b^{2} a^{2} k^{2} + 2 b^{2} a^{2} k y - b^{2} a^{2} y^{2} + b^{2} (b^{2} h^{2}) + b^{2} (- a^{2} b^{2})))}}{2 b^{2}}$

Step 5.5.1.5.1.3.4

Multiply $b^{2}$ by $b^{2}$ by adding the exponents.

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Step 5.5.1.5.1.3.4.1

Move $b^{2}$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{4} h^{2} - (- b^{2} a^{2} k^{2} + 2 b^{2} a^{2} k y - b^{2} a^{2} y^{2} + b^{2} b^{2} h^{2} + b^{2} (- a^{2} b^{2})))}}{2 b^{2}}$

Step 5.5.1.5.1.3.4.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{4} h^{2} - (- b^{2} a^{2} k^{2} + 2 b^{2} a^{2} k y - b^{2} a^{2} y^{2} + b^{2 + 2} h^{2} + b^{2} (- a^{2} b^{2})))}}{2 b^{2}}$

Step 5.5.1.5.1.3.4.3

Add $2$ and $2$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{4} h^{2} - (- b^{2} a^{2} k^{2} + 2 b^{2} a^{2} k y - b^{2} a^{2} y^{2} + b^{4} h^{2} + b^{2} (- a^{2} b^{2})))}}{2 b^{2}}$

Step 5.5.1.5.1.3.5

Rewrite using the commutative property of multiplication.

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{4} h^{2} - (- b^{2} a^{2} k^{2} + 2 b^{2} a^{2} k y - b^{2} a^{2} y^{2} + b^{4} h^{2} - b^{2} (a^{2} b^{2})))}}{2 b^{2}}$

Step 5.5.1.5.1.4

Multiply $b^{2}$ by $b^{2}$ by adding the exponents.

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Step 5.5.1.5.1.4.1

Move $b^{2}$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{4} h^{2} - (- b^{2} a^{2} k^{2} + 2 b^{2} a^{2} k y - b^{2} a^{2} y^{2} + b^{4} h^{2} - b^{2} b^{2} a^{2}))}}{2 b^{2}}$

Step 5.5.1.5.1.4.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{4} h^{2} - (- b^{2} a^{2} k^{2} + 2 b^{2} a^{2} k y - b^{2} a^{2} y^{2} + b^{4} h^{2} - b^{2 + 2} a^{2}))}}{2 b^{2}}$

Step 5.5.1.5.1.4.3

Add $2$ and $2$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{4} h^{2} - (- b^{2} a^{2} k^{2} + 2 b^{2} a^{2} k y - b^{2} a^{2} y^{2} + b^{4} h^{2} - b^{4} a^{2}))}}{2 b^{2}}$

Step 5.5.1.5.1.5

Apply the distributive property.

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{4} h^{2} - (- b^{2} a^{2} k^{2}) - (2 b^{2} a^{2} k y) - (- b^{2} a^{2} y^{2}) - (b^{4} h^{2}) - (- b^{4} a^{2}))}}{2 b^{2}}$

Step 5.5.1.5.1.6

Simplify.

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Step 5.5.1.5.1.6.1

Multiply $- (- b^{2} a^{2} k^{2})$ .

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Step 5.5.1.5.1.6.1.1

Multiply $- 1$ by $- 1$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{4} h^{2} + 1 (b^{2} a^{2} k^{2}) - (2 b^{2} a^{2} k y) - (- b^{2} a^{2} y^{2}) - (b^{4} h^{2}) - (- b^{4} a^{2}))}}{2 b^{2}}$

Step 5.5.1.5.1.6.1.2

Multiply $b^{2}$ by $1$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{4} h^{2} + b^{2} (a^{2} k^{2}) - (2 b^{2} a^{2} k y) - (- b^{2} a^{2} y^{2}) - (b^{4} h^{2}) - (- b^{4} a^{2}))}}{2 b^{2}}$

Step 5.5.1.5.1.6.2

Multiply $2$ by $- 1$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{4} h^{2} + b^{2} (a^{2} k^{2}) - 2 (b^{2} a^{2} k y) - (- b^{2} a^{2} y^{2}) - (b^{4} h^{2}) - (- b^{4} a^{2}))}}{2 b^{2}}$

Step 5.5.1.5.1.6.3

Multiply $- (- b^{2} a^{2} y^{2})$ .

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Step 5.5.1.5.1.6.3.1

Multiply $- 1$ by $- 1$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{4} h^{2} + b^{2} (a^{2} k^{2}) - 2 (b^{2} a^{2} k y) + 1 (b^{2} a^{2} y^{2}) - (b^{4} h^{2}) - (- b^{4} a^{2}))}}{2 b^{2}}$

Step 5.5.1.5.1.6.3.2

Multiply $b^{2}$ by $1$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{4} h^{2} + b^{2} (a^{2} k^{2}) - 2 (b^{2} a^{2} k y) + b^{2} (a^{2} y^{2}) - (b^{4} h^{2}) - (- b^{4} a^{2}))}}{2 b^{2}}$

Step 5.5.1.5.1.6.4

Multiply $- (- b^{4} a^{2})$ .

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Step 5.5.1.5.1.6.4.1

Multiply $- 1$ by $- 1$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{4} h^{2} + b^{2} (a^{2} k^{2}) - 2 (b^{2} a^{2} k y) + b^{2} (a^{2} y^{2}) - b^{4} h^{2} + 1 (b^{4} a^{2}))}}{2 b^{2}}$

Step 5.5.1.5.1.6.4.2

Multiply $b^{4}$ by $1$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{4} h^{2} + b^{2} (a^{2} k^{2}) - 2 (b^{2} a^{2} k y) + b^{2} (a^{2} y^{2}) - b^{4} h^{2} + b^{4} a^{2})}}{2 b^{2}}$

Step 5.5.1.5.1.7

Remove parentheses.

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{4} h^{2} + b^{2} a^{2} k^{2} - 2 b^{2} a^{2} k y + b^{2} a^{2} y^{2} - b^{4} h^{2} + b^{4} a^{2})}}{2 b^{2}}$

Step 5.5.1.5.2

Combine the opposite terms in $b^{4} h^{2} + b^{2} a^{2} k^{2} - 2 b^{2} a^{2} k y + b^{2} a^{2} y^{2} - b^{4} h^{2} + b^{4} a^{2}$ .

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Step 5.5.1.5.2.1

Subtract $b^{4} h^{2}$ from $b^{4} h^{2}$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{2} a^{2} k^{2} - 2 b^{2} a^{2} k y + b^{2} a^{2} y^{2} + 0 + b^{4} a^{2})}}{2 b^{2}}$

Step 5.5.1.5.2.2

Add $b^{2} a^{2} k^{2} - 2 b^{2} a^{2} k y + b^{2} a^{2} y^{2}$ and $0$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{2} a^{2} k^{2} - 2 b^{2} a^{2} k y + b^{2} a^{2} y^{2} + b^{4} a^{2})}}{2 b^{2}}$

Step 5.5.1.6

Factor $b^{2} a^{2}$ out of $b^{2} a^{2} k^{2} - 2 b^{2} a^{2} k y + b^{2} a^{2} y^{2} + b^{4} a^{2}$ .

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Step 5.5.1.6.1

Factor $b^{2} a^{2}$ out of $b^{2} a^{2} k^{2}$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{2} a^{2} (k^{2}) - 2 b^{2} a^{2} k y + b^{2} a^{2} y^{2} + b^{4} a^{2})}}{2 b^{2}}$

Step 5.5.1.6.2

Factor $b^{2} a^{2}$ out of $- 2 b^{2} a^{2} k y$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{2} a^{2} (k^{2}) + b^{2} a^{2} (- 2 k y) + b^{2} a^{2} y^{2} + b^{4} a^{2})}}{2 b^{2}}$

Step 5.5.1.6.3

Factor $b^{2} a^{2}$ out of $b^{2} a^{2} y^{2}$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{2} a^{2} (k^{2}) + b^{2} a^{2} (- 2 k y) + b^{2} a^{2} (y^{2}) + b^{4} a^{2})}}{2 b^{2}}$

Step 5.5.1.6.4

Factor $b^{2} a^{2}$ out of $b^{4} a^{2}$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{2} a^{2} (k^{2}) + b^{2} a^{2} (- 2 k y) + b^{2} a^{2} (y^{2}) + b^{2} a^{2} (b^{2}))}}{2 b^{2}}$

Step 5.5.1.6.5

Factor $b^{2} a^{2}$ out of $b^{2} a^{2} (k^{2}) + b^{2} a^{2} (- 2 k y)$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{2} a^{2} (k^{2} - 2 k y) + b^{2} a^{2} (y^{2}) + b^{2} a^{2} (b^{2}))}}{2 b^{2}}$

Step 5.5.1.6.6

Factor $b^{2} a^{2}$ out of $b^{2} a^{2} (k^{2} - 2 k y) + b^{2} a^{2} (y^{2})$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{2} a^{2} (k^{2} - 2 k y + y^{2}) + b^{2} a^{2} (b^{2}))}}{2 b^{2}}$

Step 5.5.1.6.7

Factor $b^{2} a^{2}$ out of $b^{2} a^{2} (k^{2} - 2 k y + y^{2}) + b^{2} a^{2} (b^{2})$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{2} a^{2} (k^{2} - 2 k y + y^{2} + b^{2}))}}{2 b^{2}}$

$x = \frac{2 b^{2} h \pm \sqrt{4 b^{2} a^{2} (k^{2} - 2 k y + y^{2} + b^{2})}}{2 b^{2}}$

Step 5.5.1.7

Rewrite $4 b^{2} a^{2} (k^{2} - 2 k y + y^{2} + b^{2})$ as ${(2 b a)}^{2} (k^{2} - 2 k y + y^{2} + b^{2})$ .

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Step 5.5.1.7.1

Rewrite $4$ as $2^{2}$ .

$x = \frac{2 b^{2} h \pm \sqrt{2^{2} b^{2} a^{2} (k^{2} - 2 k y + y^{2} + b^{2})}}{2 b^{2}}$

Step 5.5.1.7.2

Rewrite $2^{2} b^{2} a^{2}$ as ${(2 b a)}^{2}$ .

$x = \frac{2 b^{2} h \pm \sqrt{{(2 b a)}^{2} (k^{2} - 2 k y + y^{2} + b^{2})}}{2 b^{2}}$

Step 5.5.1.8

Pull terms out from under the radical.

$x = \frac{2 b^{2} h \pm 2 b a \sqrt{k^{2} - 2 k y + y^{2} + b^{2}}}{2 b^{2}}$

Step 5.5.2

Simplify $\frac{2 b^{2} h \pm 2 b a \sqrt{k^{2} - 2 k y + y^{2} + b^{2}}}{2 b^{2}}$ .

$x = \frac{b h \pm a \sqrt{k^{2} - 2 k y + y^{2} + b^{2}}}{b}$

Step 5.6

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 5.6.1

Simplify the numerator.

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Step 5.6.1.1

Rewrite $a^{2} b^{2}$ as ${(a b)}^{2}$ .

$x = \frac{2 b^{2} h \pm \sqrt{{(- 2 (b^{2} h))}^{2} - 4 \cdot (b^{2} \cdot (- {(a k)}^{2} + 2 (a^{2} (k y)) - {(a y)}^{2} + {(b h)}^{2} - {(a b)}^{2}))}}{2 b^{2}}$

Step 5.6.1.2

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Step 5.6.1.2.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 5.6.1.2.1.1

Apply the product rule to $- 2 b^{2} h$ .

$x = \frac{2 b^{2} h \pm \sqrt{{(- 2 b^{2})}^{2} h^{2} - 4 \cdot u}}{2 b^{2}}$

Step 5.6.1.2.1.2

Apply the product rule to $- 2 b^{2}$ .

$x = \frac{2 b^{2} h \pm \sqrt{{(- 2)}^{2} {(b^{2})}^{2} h^{2} - 4 \cdot u}}{2 b^{2}}$

Step 5.6.1.2.2

Raise $- 2$ to the power of $2$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 {(b^{2})}^{2} h^{2} - 4 \cdot u}}{2 b^{2}}$

Step 5.6.1.2.3

Multiply the exponents in ${(b^{2})}^{2}$ .

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Step 5.6.1.2.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 b^{2 \cdot 2} h^{2} - 4 \cdot u}}{2 b^{2}}$

Step 5.6.1.2.3.2

Multiply $2$ by $2$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 b^{4} h^{2} - 4 \cdot u}}{2 b^{2}}$

$x = \frac{2 b^{2} h \pm \sqrt{4 b^{4} h^{2} - 4 u}}{2 b^{2}}$

Step 5.6.1.3

Factor $4$ out of $4 b^{4} h^{2} - 4 u$ .

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Step 5.6.1.3.1

Factor $4$ out of $4 b^{4} h^{2}$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{4} h^{2}) - 4 u}}{2 b^{2}}$

Step 5.6.1.3.2

Factor $4$ out of $- 4 u$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{4} h^{2}) + 4 (- u)}}{2 b^{2}}$

Step 5.6.1.3.3

Factor $4$ out of $4 (b^{4} h^{2}) + 4 (- u)$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{4} h^{2} - u)}}{2 b^{2}}$

Step 5.6.1.4

Replace all occurrences of $u$ with $b^{2} \cdot (- {(a k)}^{2} + 2 (a^{2} (k y)) - {(a y)}^{2} + {(b h)}^{2} - {(a b)}^{2})$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{4} h^{2} - (b^{2} \cdot (- {(a k)}^{2} + 2 (a^{2} (k y)) - {(a y)}^{2} + {(b h)}^{2} - {(a b)}^{2})))}}{2 b^{2}}$

Step 5.6.1.5

Simplify.

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Step 5.6.1.5.1

Simplify each term.

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Step 5.6.1.5.1.1

Simplify each term.

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Step 5.6.1.5.1.1.1

Apply the product rule to $a k$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{4} h^{2} - (b^{2} \cdot (- (a^{2} k^{2}) + 2 (a^{2} (k y)) - {(a y)}^{2} + {(b h)}^{2} - {(a b)}^{2})))}}{2 b^{2}}$

Step 5.6.1.5.1.1.2

Apply the product rule to $a y$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{4} h^{2} - (b^{2} \cdot (- a^{2} k^{2} + 2 a^{2} k y - (a^{2} y^{2}) + {(b h)}^{2} - {(a b)}^{2})))}}{2 b^{2}}$

Step 5.6.1.5.1.1.3

Apply the product rule to $b h$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{4} h^{2} - (b^{2} \cdot (- a^{2} k^{2} + 2 a^{2} k y - a^{2} y^{2} + b^{2} h^{2} - {(a b)}^{2})))}}{2 b^{2}}$

Step 5.6.1.5.1.1.4

Apply the product rule to $a b$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{4} h^{2} - (b^{2} \cdot (- a^{2} k^{2} + 2 a^{2} k y - a^{2} y^{2} + b^{2} h^{2} - a^{2} b^{2})))}}{2 b^{2}}$

Step 5.6.1.5.1.2

Apply the distributive property.

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{4} h^{2} - (b^{2} (- a^{2} k^{2}) + b^{2} (2 a^{2} k y) + b^{2} (- a^{2} y^{2}) + b^{2} (b^{2} h^{2}) + b^{2} (- a^{2} b^{2})))}}{2 b^{2}}$

Step 5.6.1.5.1.3

Simplify.

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Step 5.6.1.5.1.3.1

Rewrite using the commutative property of multiplication.

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{4} h^{2} - (- b^{2} a^{2} k^{2} + b^{2} (2 a^{2} k y) + b^{2} (- a^{2} y^{2}) + b^{2} (b^{2} h^{2}) + b^{2} (- a^{2} b^{2})))}}{2 b^{2}}$

Step 5.6.1.5.1.3.2

Rewrite using the commutative property of multiplication.

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{4} h^{2} - (- b^{2} a^{2} k^{2} + 2 b^{2} a^{2} k y + b^{2} (- a^{2} y^{2}) + b^{2} (b^{2} h^{2}) + b^{2} (- a^{2} b^{2})))}}{2 b^{2}}$

Step 5.6.1.5.1.3.3

Rewrite using the commutative property of multiplication.

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{4} h^{2} - (- b^{2} a^{2} k^{2} + 2 b^{2} a^{2} k y - b^{2} a^{2} y^{2} + b^{2} (b^{2} h^{2}) + b^{2} (- a^{2} b^{2})))}}{2 b^{2}}$

Step 5.6.1.5.1.3.4

Multiply $b^{2}$ by $b^{2}$ by adding the exponents.

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Step 5.6.1.5.1.3.4.1

Move $b^{2}$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{4} h^{2} - (- b^{2} a^{2} k^{2} + 2 b^{2} a^{2} k y - b^{2} a^{2} y^{2} + b^{2} b^{2} h^{2} + b^{2} (- a^{2} b^{2})))}}{2 b^{2}}$

Step 5.6.1.5.1.3.4.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{4} h^{2} - (- b^{2} a^{2} k^{2} + 2 b^{2} a^{2} k y - b^{2} a^{2} y^{2} + b^{2 + 2} h^{2} + b^{2} (- a^{2} b^{2})))}}{2 b^{2}}$

Step 5.6.1.5.1.3.4.3

Add $2$ and $2$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{4} h^{2} - (- b^{2} a^{2} k^{2} + 2 b^{2} a^{2} k y - b^{2} a^{2} y^{2} + b^{4} h^{2} + b^{2} (- a^{2} b^{2})))}}{2 b^{2}}$

Step 5.6.1.5.1.3.5

Rewrite using the commutative property of multiplication.

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{4} h^{2} - (- b^{2} a^{2} k^{2} + 2 b^{2} a^{2} k y - b^{2} a^{2} y^{2} + b^{4} h^{2} - b^{2} (a^{2} b^{2})))}}{2 b^{2}}$

Step 5.6.1.5.1.4

Multiply $b^{2}$ by $b^{2}$ by adding the exponents.

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Step 5.6.1.5.1.4.1

Move $b^{2}$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{4} h^{2} - (- b^{2} a^{2} k^{2} + 2 b^{2} a^{2} k y - b^{2} a^{2} y^{2} + b^{4} h^{2} - b^{2} b^{2} a^{2}))}}{2 b^{2}}$

Step 5.6.1.5.1.4.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{4} h^{2} - (- b^{2} a^{2} k^{2} + 2 b^{2} a^{2} k y - b^{2} a^{2} y^{2} + b^{4} h^{2} - b^{2 + 2} a^{2}))}}{2 b^{2}}$

Step 5.6.1.5.1.4.3

Add $2$ and $2$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{4} h^{2} - (- b^{2} a^{2} k^{2} + 2 b^{2} a^{2} k y - b^{2} a^{2} y^{2} + b^{4} h^{2} - b^{4} a^{2}))}}{2 b^{2}}$

Step 5.6.1.5.1.5

Apply the distributive property.

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{4} h^{2} - (- b^{2} a^{2} k^{2}) - (2 b^{2} a^{2} k y) - (- b^{2} a^{2} y^{2}) - (b^{4} h^{2}) - (- b^{4} a^{2}))}}{2 b^{2}}$

Step 5.6.1.5.1.6

Simplify.

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Step 5.6.1.5.1.6.1

Multiply $- (- b^{2} a^{2} k^{2})$ .

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Step 5.6.1.5.1.6.1.1

Multiply $- 1$ by $- 1$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{4} h^{2} + 1 (b^{2} a^{2} k^{2}) - (2 b^{2} a^{2} k y) - (- b^{2} a^{2} y^{2}) - (b^{4} h^{2}) - (- b^{4} a^{2}))}}{2 b^{2}}$

Step 5.6.1.5.1.6.1.2

Multiply $b^{2}$ by $1$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{4} h^{2} + b^{2} (a^{2} k^{2}) - (2 b^{2} a^{2} k y) - (- b^{2} a^{2} y^{2}) - (b^{4} h^{2}) - (- b^{4} a^{2}))}}{2 b^{2}}$

Step 5.6.1.5.1.6.2

Multiply $2$ by $- 1$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{4} h^{2} + b^{2} (a^{2} k^{2}) - 2 (b^{2} a^{2} k y) - (- b^{2} a^{2} y^{2}) - (b^{4} h^{2}) - (- b^{4} a^{2}))}}{2 b^{2}}$

Step 5.6.1.5.1.6.3

Multiply $- (- b^{2} a^{2} y^{2})$ .

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Step 5.6.1.5.1.6.3.1

Multiply $- 1$ by $- 1$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{4} h^{2} + b^{2} (a^{2} k^{2}) - 2 (b^{2} a^{2} k y) + 1 (b^{2} a^{2} y^{2}) - (b^{4} h^{2}) - (- b^{4} a^{2}))}}{2 b^{2}}$

Step 5.6.1.5.1.6.3.2

Multiply $b^{2}$ by $1$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{4} h^{2} + b^{2} (a^{2} k^{2}) - 2 (b^{2} a^{2} k y) + b^{2} (a^{2} y^{2}) - (b^{4} h^{2}) - (- b^{4} a^{2}))}}{2 b^{2}}$

Step 5.6.1.5.1.6.4

Multiply $- (- b^{4} a^{2})$ .

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Step 5.6.1.5.1.6.4.1

Multiply $- 1$ by $- 1$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{4} h^{2} + b^{2} (a^{2} k^{2}) - 2 (b^{2} a^{2} k y) + b^{2} (a^{2} y^{2}) - b^{4} h^{2} + 1 (b^{4} a^{2}))}}{2 b^{2}}$

Step 5.6.1.5.1.6.4.2

Multiply $b^{4}$ by $1$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{4} h^{2} + b^{2} (a^{2} k^{2}) - 2 (b^{2} a^{2} k y) + b^{2} (a^{2} y^{2}) - b^{4} h^{2} + b^{4} a^{2})}}{2 b^{2}}$

Step 5.6.1.5.1.7

Remove parentheses.

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{4} h^{2} + b^{2} a^{2} k^{2} - 2 b^{2} a^{2} k y + b^{2} a^{2} y^{2} - b^{4} h^{2} + b^{4} a^{2})}}{2 b^{2}}$

Step 5.6.1.5.2

Combine the opposite terms in $b^{4} h^{2} + b^{2} a^{2} k^{2} - 2 b^{2} a^{2} k y + b^{2} a^{2} y^{2} - b^{4} h^{2} + b^{4} a^{2}$ .

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Step 5.6.1.5.2.1

Subtract $b^{4} h^{2}$ from $b^{4} h^{2}$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{2} a^{2} k^{2} - 2 b^{2} a^{2} k y + b^{2} a^{2} y^{2} + 0 + b^{4} a^{2})}}{2 b^{2}}$

Step 5.6.1.5.2.2

Add $b^{2} a^{2} k^{2} - 2 b^{2} a^{2} k y + b^{2} a^{2} y^{2}$ and $0$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{2} a^{2} k^{2} - 2 b^{2} a^{2} k y + b^{2} a^{2} y^{2} + b^{4} a^{2})}}{2 b^{2}}$

Step 5.6.1.6

Factor $b^{2} a^{2}$ out of $b^{2} a^{2} k^{2} - 2 b^{2} a^{2} k y + b^{2} a^{2} y^{2} + b^{4} a^{2}$ .

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Step 5.6.1.6.1

Factor $b^{2} a^{2}$ out of $b^{2} a^{2} k^{2}$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{2} a^{2} (k^{2}) - 2 b^{2} a^{2} k y + b^{2} a^{2} y^{2} + b^{4} a^{2})}}{2 b^{2}}$

Step 5.6.1.6.2

Factor $b^{2} a^{2}$ out of $- 2 b^{2} a^{2} k y$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{2} a^{2} (k^{2}) + b^{2} a^{2} (- 2 k y) + b^{2} a^{2} y^{2} + b^{4} a^{2})}}{2 b^{2}}$

Step 5.6.1.6.3

Factor $b^{2} a^{2}$ out of $b^{2} a^{2} y^{2}$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{2} a^{2} (k^{2}) + b^{2} a^{2} (- 2 k y) + b^{2} a^{2} (y^{2}) + b^{4} a^{2})}}{2 b^{2}}$

Step 5.6.1.6.4

Factor $b^{2} a^{2}$ out of $b^{4} a^{2}$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{2} a^{2} (k^{2}) + b^{2} a^{2} (- 2 k y) + b^{2} a^{2} (y^{2}) + b^{2} a^{2} (b^{2}))}}{2 b^{2}}$

Step 5.6.1.6.5

Factor $b^{2} a^{2}$ out of $b^{2} a^{2} (k^{2}) + b^{2} a^{2} (- 2 k y)$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{2} a^{2} (k^{2} - 2 k y) + b^{2} a^{2} (y^{2}) + b^{2} a^{2} (b^{2}))}}{2 b^{2}}$

Step 5.6.1.6.6

Factor $b^{2} a^{2}$ out of $b^{2} a^{2} (k^{2} - 2 k y) + b^{2} a^{2} (y^{2})$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{2} a^{2} (k^{2} - 2 k y + y^{2}) + b^{2} a^{2} (b^{2}))}}{2 b^{2}}$

Step 5.6.1.6.7

Factor $b^{2} a^{2}$ out of $b^{2} a^{2} (k^{2} - 2 k y + y^{2}) + b^{2} a^{2} (b^{2})$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{2} a^{2} (k^{2} - 2 k y + y^{2} + b^{2}))}}{2 b^{2}}$

$x = \frac{2 b^{2} h \pm \sqrt{4 b^{2} a^{2} (k^{2} - 2 k y + y^{2} + b^{2})}}{2 b^{2}}$

Step 5.6.1.7

Rewrite $4 b^{2} a^{2} (k^{2} - 2 k y + y^{2} + b^{2})$ as ${(2 b a)}^{2} (k^{2} - 2 k y + y^{2} + b^{2})$ .

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Step 5.6.1.7.1

Rewrite $4$ as $2^{2}$ .

$x = \frac{2 b^{2} h \pm \sqrt{2^{2} b^{2} a^{2} (k^{2} - 2 k y + y^{2} + b^{2})}}{2 b^{2}}$

Step 5.6.1.7.2

Rewrite $2^{2} b^{2} a^{2}$ as ${(2 b a)}^{2}$ .

$x = \frac{2 b^{2} h \pm \sqrt{{(2 b a)}^{2} (k^{2} - 2 k y + y^{2} + b^{2})}}{2 b^{2}}$

Step 5.6.1.8

Pull terms out from under the radical.

$x = \frac{2 b^{2} h \pm 2 b a \sqrt{k^{2} - 2 k y + y^{2} + b^{2}}}{2 b^{2}}$

Step 5.6.2

Simplify $\frac{2 b^{2} h \pm 2 b a \sqrt{k^{2} - 2 k y + y^{2} + b^{2}}}{2 b^{2}}$ .

$x = \frac{b h \pm a \sqrt{k^{2} - 2 k y + y^{2} + b^{2}}}{b}$

Step 5.6.3

Change the $\pm$ to $+$ .

$x = \frac{b h + a \sqrt{k^{2} - 2 k y + y^{2} + b^{2}}}{b}$

Step 5.7

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 5.7.1

Simplify the numerator.

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Step 5.7.1.1

Rewrite $a^{2} b^{2}$ as ${(a b)}^{2}$ .

$x = \frac{2 b^{2} h \pm \sqrt{{(- 2 (b^{2} h))}^{2} - 4 \cdot (b^{2} \cdot (- {(a k)}^{2} + 2 (a^{2} (k y)) - {(a y)}^{2} + {(b h)}^{2} - {(a b)}^{2}))}}{2 b^{2}}$

Step 5.7.1.2

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Step 5.7.1.2.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 5.7.1.2.1.1

Apply the product rule to $- 2 b^{2} h$ .

$x = \frac{2 b^{2} h \pm \sqrt{{(- 2 b^{2})}^{2} h^{2} - 4 \cdot u}}{2 b^{2}}$

Step 5.7.1.2.1.2

Apply the product rule to $- 2 b^{2}$ .

$x = \frac{2 b^{2} h \pm \sqrt{{(- 2)}^{2} {(b^{2})}^{2} h^{2} - 4 \cdot u}}{2 b^{2}}$

Step 5.7.1.2.2

Raise $- 2$ to the power of $2$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 {(b^{2})}^{2} h^{2} - 4 \cdot u}}{2 b^{2}}$

Step 5.7.1.2.3

Multiply the exponents in ${(b^{2})}^{2}$ .

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Step 5.7.1.2.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 b^{2 \cdot 2} h^{2} - 4 \cdot u}}{2 b^{2}}$

Step 5.7.1.2.3.2

Multiply $2$ by $2$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 b^{4} h^{2} - 4 \cdot u}}{2 b^{2}}$

$x = \frac{2 b^{2} h \pm \sqrt{4 b^{4} h^{2} - 4 u}}{2 b^{2}}$

Step 5.7.1.3

Factor $4$ out of $4 b^{4} h^{2} - 4 u$ .

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Step 5.7.1.3.1

Factor $4$ out of $4 b^{4} h^{2}$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{4} h^{2}) - 4 u}}{2 b^{2}}$

Step 5.7.1.3.2

Factor $4$ out of $- 4 u$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{4} h^{2}) + 4 (- u)}}{2 b^{2}}$

Step 5.7.1.3.3

Factor $4$ out of $4 (b^{4} h^{2}) + 4 (- u)$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{4} h^{2} - u)}}{2 b^{2}}$

Step 5.7.1.4

Replace all occurrences of $u$ with $b^{2} \cdot (- {(a k)}^{2} + 2 (a^{2} (k y)) - {(a y)}^{2} + {(b h)}^{2} - {(a b)}^{2})$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{4} h^{2} - (b^{2} \cdot (- {(a k)}^{2} + 2 (a^{2} (k y)) - {(a y)}^{2} + {(b h)}^{2} - {(a b)}^{2})))}}{2 b^{2}}$

Step 5.7.1.5

Simplify.

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Step 5.7.1.5.1

Simplify each term.

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Step 5.7.1.5.1.1

Simplify each term.

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Step 5.7.1.5.1.1.1

Apply the product rule to $a k$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{4} h^{2} - (b^{2} \cdot (- (a^{2} k^{2}) + 2 (a^{2} (k y)) - {(a y)}^{2} + {(b h)}^{2} - {(a b)}^{2})))}}{2 b^{2}}$

Step 5.7.1.5.1.1.2

Apply the product rule to $a y$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{4} h^{2} - (b^{2} \cdot (- a^{2} k^{2} + 2 a^{2} k y - (a^{2} y^{2}) + {(b h)}^{2} - {(a b)}^{2})))}}{2 b^{2}}$

Step 5.7.1.5.1.1.3

Apply the product rule to $b h$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{4} h^{2} - (b^{2} \cdot (- a^{2} k^{2} + 2 a^{2} k y - a^{2} y^{2} + b^{2} h^{2} - {(a b)}^{2})))}}{2 b^{2}}$

Step 5.7.1.5.1.1.4

Apply the product rule to $a b$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{4} h^{2} - (b^{2} \cdot (- a^{2} k^{2} + 2 a^{2} k y - a^{2} y^{2} + b^{2} h^{2} - a^{2} b^{2})))}}{2 b^{2}}$

Step 5.7.1.5.1.2

Apply the distributive property.

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{4} h^{2} - (b^{2} (- a^{2} k^{2}) + b^{2} (2 a^{2} k y) + b^{2} (- a^{2} y^{2}) + b^{2} (b^{2} h^{2}) + b^{2} (- a^{2} b^{2})))}}{2 b^{2}}$

Step 5.7.1.5.1.3

Simplify.

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Step 5.7.1.5.1.3.1

Rewrite using the commutative property of multiplication.

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{4} h^{2} - (- b^{2} a^{2} k^{2} + b^{2} (2 a^{2} k y) + b^{2} (- a^{2} y^{2}) + b^{2} (b^{2} h^{2}) + b^{2} (- a^{2} b^{2})))}}{2 b^{2}}$

Step 5.7.1.5.1.3.2

Rewrite using the commutative property of multiplication.

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{4} h^{2} - (- b^{2} a^{2} k^{2} + 2 b^{2} a^{2} k y + b^{2} (- a^{2} y^{2}) + b^{2} (b^{2} h^{2}) + b^{2} (- a^{2} b^{2})))}}{2 b^{2}}$

Step 5.7.1.5.1.3.3

Rewrite using the commutative property of multiplication.

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{4} h^{2} - (- b^{2} a^{2} k^{2} + 2 b^{2} a^{2} k y - b^{2} a^{2} y^{2} + b^{2} (b^{2} h^{2}) + b^{2} (- a^{2} b^{2})))}}{2 b^{2}}$

Step 5.7.1.5.1.3.4

Multiply $b^{2}$ by $b^{2}$ by adding the exponents.

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Step 5.7.1.5.1.3.4.1

Move $b^{2}$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{4} h^{2} - (- b^{2} a^{2} k^{2} + 2 b^{2} a^{2} k y - b^{2} a^{2} y^{2} + b^{2} b^{2} h^{2} + b^{2} (- a^{2} b^{2})))}}{2 b^{2}}$

Step 5.7.1.5.1.3.4.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{4} h^{2} - (- b^{2} a^{2} k^{2} + 2 b^{2} a^{2} k y - b^{2} a^{2} y^{2} + b^{2 + 2} h^{2} + b^{2} (- a^{2} b^{2})))}}{2 b^{2}}$

Step 5.7.1.5.1.3.4.3

Add $2$ and $2$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{4} h^{2} - (- b^{2} a^{2} k^{2} + 2 b^{2} a^{2} k y - b^{2} a^{2} y^{2} + b^{4} h^{2} + b^{2} (- a^{2} b^{2})))}}{2 b^{2}}$

Step 5.7.1.5.1.3.5

Rewrite using the commutative property of multiplication.

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{4} h^{2} - (- b^{2} a^{2} k^{2} + 2 b^{2} a^{2} k y - b^{2} a^{2} y^{2} + b^{4} h^{2} - b^{2} (a^{2} b^{2})))}}{2 b^{2}}$

Step 5.7.1.5.1.4

Multiply $b^{2}$ by $b^{2}$ by adding the exponents.

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Step 5.7.1.5.1.4.1

Move $b^{2}$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{4} h^{2} - (- b^{2} a^{2} k^{2} + 2 b^{2} a^{2} k y - b^{2} a^{2} y^{2} + b^{4} h^{2} - b^{2} b^{2} a^{2}))}}{2 b^{2}}$

Step 5.7.1.5.1.4.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{4} h^{2} - (- b^{2} a^{2} k^{2} + 2 b^{2} a^{2} k y - b^{2} a^{2} y^{2} + b^{4} h^{2} - b^{2 + 2} a^{2}))}}{2 b^{2}}$

Step 5.7.1.5.1.4.3

Add $2$ and $2$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{4} h^{2} - (- b^{2} a^{2} k^{2} + 2 b^{2} a^{2} k y - b^{2} a^{2} y^{2} + b^{4} h^{2} - b^{4} a^{2}))}}{2 b^{2}}$

Step 5.7.1.5.1.5

Apply the distributive property.

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{4} h^{2} - (- b^{2} a^{2} k^{2}) - (2 b^{2} a^{2} k y) - (- b^{2} a^{2} y^{2}) - (b^{4} h^{2}) - (- b^{4} a^{2}))}}{2 b^{2}}$

Step 5.7.1.5.1.6

Simplify.

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Step 5.7.1.5.1.6.1

Multiply $- (- b^{2} a^{2} k^{2})$ .

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Step 5.7.1.5.1.6.1.1

Multiply $- 1$ by $- 1$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{4} h^{2} + 1 (b^{2} a^{2} k^{2}) - (2 b^{2} a^{2} k y) - (- b^{2} a^{2} y^{2}) - (b^{4} h^{2}) - (- b^{4} a^{2}))}}{2 b^{2}}$

Step 5.7.1.5.1.6.1.2

Multiply $b^{2}$ by $1$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{4} h^{2} + b^{2} (a^{2} k^{2}) - (2 b^{2} a^{2} k y) - (- b^{2} a^{2} y^{2}) - (b^{4} h^{2}) - (- b^{4} a^{2}))}}{2 b^{2}}$

Step 5.7.1.5.1.6.2

Multiply $2$ by $- 1$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{4} h^{2} + b^{2} (a^{2} k^{2}) - 2 (b^{2} a^{2} k y) - (- b^{2} a^{2} y^{2}) - (b^{4} h^{2}) - (- b^{4} a^{2}))}}{2 b^{2}}$

Step 5.7.1.5.1.6.3

Multiply $- (- b^{2} a^{2} y^{2})$ .

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Step 5.7.1.5.1.6.3.1

Multiply $- 1$ by $- 1$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{4} h^{2} + b^{2} (a^{2} k^{2}) - 2 (b^{2} a^{2} k y) + 1 (b^{2} a^{2} y^{2}) - (b^{4} h^{2}) - (- b^{4} a^{2}))}}{2 b^{2}}$

Step 5.7.1.5.1.6.3.2

Multiply $b^{2}$ by $1$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{4} h^{2} + b^{2} (a^{2} k^{2}) - 2 (b^{2} a^{2} k y) + b^{2} (a^{2} y^{2}) - (b^{4} h^{2}) - (- b^{4} a^{2}))}}{2 b^{2}}$

Step 5.7.1.5.1.6.4

Multiply $- (- b^{4} a^{2})$ .

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Step 5.7.1.5.1.6.4.1

Multiply $- 1$ by $- 1$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{4} h^{2} + b^{2} (a^{2} k^{2}) - 2 (b^{2} a^{2} k y) + b^{2} (a^{2} y^{2}) - b^{4} h^{2} + 1 (b^{4} a^{2}))}}{2 b^{2}}$

Step 5.7.1.5.1.6.4.2

Multiply $b^{4}$ by $1$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{4} h^{2} + b^{2} (a^{2} k^{2}) - 2 (b^{2} a^{2} k y) + b^{2} (a^{2} y^{2}) - b^{4} h^{2} + b^{4} a^{2})}}{2 b^{2}}$

Step 5.7.1.5.1.7

Remove parentheses.

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{4} h^{2} + b^{2} a^{2} k^{2} - 2 b^{2} a^{2} k y + b^{2} a^{2} y^{2} - b^{4} h^{2} + b^{4} a^{2})}}{2 b^{2}}$

Step 5.7.1.5.2

Combine the opposite terms in $b^{4} h^{2} + b^{2} a^{2} k^{2} - 2 b^{2} a^{2} k y + b^{2} a^{2} y^{2} - b^{4} h^{2} + b^{4} a^{2}$ .

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Step 5.7.1.5.2.1

Subtract $b^{4} h^{2}$ from $b^{4} h^{2}$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{2} a^{2} k^{2} - 2 b^{2} a^{2} k y + b^{2} a^{2} y^{2} + 0 + b^{4} a^{2})}}{2 b^{2}}$

Step 5.7.1.5.2.2

Add $b^{2} a^{2} k^{2} - 2 b^{2} a^{2} k y + b^{2} a^{2} y^{2}$ and $0$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{2} a^{2} k^{2} - 2 b^{2} a^{2} k y + b^{2} a^{2} y^{2} + b^{4} a^{2})}}{2 b^{2}}$

Step 5.7.1.6

Factor $b^{2} a^{2}$ out of $b^{2} a^{2} k^{2} - 2 b^{2} a^{2} k y + b^{2} a^{2} y^{2} + b^{4} a^{2}$ .

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Step 5.7.1.6.1

Factor $b^{2} a^{2}$ out of $b^{2} a^{2} k^{2}$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{2} a^{2} (k^{2}) - 2 b^{2} a^{2} k y + b^{2} a^{2} y^{2} + b^{4} a^{2})}}{2 b^{2}}$

Step 5.7.1.6.2

Factor $b^{2} a^{2}$ out of $- 2 b^{2} a^{2} k y$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{2} a^{2} (k^{2}) + b^{2} a^{2} (- 2 k y) + b^{2} a^{2} y^{2} + b^{4} a^{2})}}{2 b^{2}}$

Step 5.7.1.6.3

Factor $b^{2} a^{2}$ out of $b^{2} a^{2} y^{2}$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{2} a^{2} (k^{2}) + b^{2} a^{2} (- 2 k y) + b^{2} a^{2} (y^{2}) + b^{4} a^{2})}}{2 b^{2}}$

Step 5.7.1.6.4

Factor $b^{2} a^{2}$ out of $b^{4} a^{2}$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{2} a^{2} (k^{2}) + b^{2} a^{2} (- 2 k y) + b^{2} a^{2} (y^{2}) + b^{2} a^{2} (b^{2}))}}{2 b^{2}}$

Step 5.7.1.6.5

Factor $b^{2} a^{2}$ out of $b^{2} a^{2} (k^{2}) + b^{2} a^{2} (- 2 k y)$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{2} a^{2} (k^{2} - 2 k y) + b^{2} a^{2} (y^{2}) + b^{2} a^{2} (b^{2}))}}{2 b^{2}}$

Step 5.7.1.6.6

Factor $b^{2} a^{2}$ out of $b^{2} a^{2} (k^{2} - 2 k y) + b^{2} a^{2} (y^{2})$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{2} a^{2} (k^{2} - 2 k y + y^{2}) + b^{2} a^{2} (b^{2}))}}{2 b^{2}}$

Step 5.7.1.6.7

Factor $b^{2} a^{2}$ out of $b^{2} a^{2} (k^{2} - 2 k y + y^{2}) + b^{2} a^{2} (b^{2})$ .

$x = \frac{2 b^{2} h \pm \sqrt{4 (b^{2} a^{2} (k^{2} - 2 k y + y^{2} + b^{2}))}}{2 b^{2}}$

$x = \frac{2 b^{2} h \pm \sqrt{4 b^{2} a^{2} (k^{2} - 2 k y + y^{2} + b^{2})}}{2 b^{2}}$

Step 5.7.1.7

Rewrite $4 b^{2} a^{2} (k^{2} - 2 k y + y^{2} + b^{2})$ as ${(2 b a)}^{2} (k^{2} - 2 k y + y^{2} + b^{2})$ .

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Step 5.7.1.7.1

Rewrite $4$ as $2^{2}$ .

$x = \frac{2 b^{2} h \pm \sqrt{2^{2} b^{2} a^{2} (k^{2} - 2 k y + y^{2} + b^{2})}}{2 b^{2}}$

Step 5.7.1.7.2

Rewrite $2^{2} b^{2} a^{2}$ as ${(2 b a)}^{2}$ .

$x = \frac{2 b^{2} h \pm \sqrt{{(2 b a)}^{2} (k^{2} - 2 k y + y^{2} + b^{2})}}{2 b^{2}}$

Step 5.7.1.8

Pull terms out from under the radical.

$x = \frac{2 b^{2} h \pm 2 b a \sqrt{k^{2} - 2 k y + y^{2} + b^{2}}}{2 b^{2}}$

Step 5.7.2

Simplify $\frac{2 b^{2} h \pm 2 b a \sqrt{k^{2} - 2 k y + y^{2} + b^{2}}}{2 b^{2}}$ .

$x = \frac{b h \pm a \sqrt{k^{2} - 2 k y + y^{2} + b^{2}}}{b}$

Step 5.7.3

Change the $\pm$ to $-$ .

$x = \frac{b h - a \sqrt{k^{2} - 2 k y + y^{2} + b^{2}}}{b}$

Step 5.8

The final answer is the combination of both solutions.

$x = \frac{b h + a \sqrt{k^{2} - 2 k y + y^{2} + b^{2}}}{b}$

$x = \frac{b h - a \sqrt{k^{2} - 2 k y + y^{2} + b^{2}}}{b}$

$x = \frac{b h + a \sqrt{k^{2} - 2 k y + y^{2} + b^{2}}}{b}$

$x = \frac{b h - a \sqrt{k^{2} - 2 k y + y^{2} + b^{2}}}{b}$