Algebra Examples

$x^{2} + {(y - 3 \sqrt{2} x)}^{2} = 1$

Step 1

Rewrite ${(y - 3 \sqrt{2} x)}^{2}$ as $(y - 3 \sqrt{2} x) (y - 3 \sqrt{2} x)$ .

$x^{2} + (y - 3 \sqrt{2} x) (y - 3 \sqrt{2} x) = 1$

Step 2

Expand $(y - 3 \sqrt{2} x) (y - 3 \sqrt{2} x)$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$x^{2} + y (y - 3 \sqrt{2} x) - 3 \sqrt{2} x (y - 3 \sqrt{2} x) = 1$

Step 2.2

Apply the distributive property.

$x^{2} + y \cdot y + y (- 3 \sqrt{2} x) - 3 \sqrt{2} x (y - 3 \sqrt{2} x) = 1$

Step 2.3

Apply the distributive property.

$x^{2} + y \cdot y + y (- 3 \sqrt{2} x) - 3 \sqrt{2} x y - 3 \sqrt{2} x (- 3 \sqrt{2} x) = 1$

Step 3

Simplify each term.

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Step 3.1

Multiply $y$ by $y$ .

$x^{2} + y^{2} + y (- 3 \sqrt{2} x) - 3 \sqrt{2} x y - 3 \sqrt{2} x (- 3 \sqrt{2} x) = 1$

Step 3.2

Rewrite using the commutative property of multiplication.

$x^{2} + y^{2} - 3 \sqrt{2} y x - 3 \sqrt{2} x y - 3 \sqrt{2} x (- 3 \sqrt{2} x) = 1$

Step 3.3

Rewrite using the commutative property of multiplication.

$x^{2} + y^{2} - 3 \sqrt{2} y x - 3 \sqrt{2} x y - 3 \sqrt{2} \cdot (- 3 \sqrt{2} x \cdot x) = 1$

Step 3.4

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

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Step 3.4.1

Move $x$ .

$x^{2} + y^{2} - 3 \sqrt{2} y x - 3 \sqrt{2} x y - 3 \sqrt{2} \cdot (- 3 \sqrt{2} (x \cdot x)) = 1$

Step 3.4.2

Multiply $x$ by $x$ .

$x^{2} + y^{2} - 3 \sqrt{2} y x - 3 \sqrt{2} x y - 3 \sqrt{2} \cdot (- 3 \sqrt{2} x^{2}) = 1$

Step 3.5

Multiply $- 3$ by $- 3$ .

$x^{2} + y^{2} - 3 \sqrt{2} y x - 3 \sqrt{2} x y + 9 \sqrt{2} \sqrt{2} x^{2} = 1$

Step 3.6

Multiply $9 \sqrt{2} \sqrt{2}$ .

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Step 3.6.1

Raise $\sqrt{2}$ to the power of $1$ .

$x^{2} + y^{2} - 3 \sqrt{2} y x - 3 \sqrt{2} x y + 9 (\sqrt{2} \sqrt{2}) x^{2} = 1$

Step 3.6.2

Raise $\sqrt{2}$ to the power of $1$ .

$x^{2} + y^{2} - 3 \sqrt{2} y x - 3 \sqrt{2} x y + 9 (\sqrt{2} \sqrt{2}) x^{2} = 1$

Step 3.6.3

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

$x^{2} + y^{2} - 3 \sqrt{2} y x - 3 \sqrt{2} x y + 9 {\sqrt{2}}^{1 + 1} x^{2} = 1$

Step 3.6.4

Add $1$ and $1$ .

$x^{2} + y^{2} - 3 \sqrt{2} y x - 3 \sqrt{2} x y + 9 {\sqrt{2}}^{2} x^{2} = 1$

Step 3.7

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 3.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$x^{2} + y^{2} - 3 \sqrt{2} y x - 3 \sqrt{2} x y + 9 {(2^{\frac{1}{2}})}^{2} x^{2} = 1$

Step 3.7.2

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

$x^{2} + y^{2} - 3 \sqrt{2} y x - 3 \sqrt{2} x y + 9 \cdot (2^{\frac{1}{2} \cdot 2} x^{2}) = 1$

Step 3.7.3

Combine $\frac{1}{2}$ and $2$ .

$x^{2} + y^{2} - 3 \sqrt{2} y x - 3 \sqrt{2} x y + 9 \cdot (2^{\frac{2}{2}} x^{2}) = 1$

Step 3.7.4

Cancel the common factor of $2$ .

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Step 3.7.4.1

Cancel the common factor.

$x^{2} + y^{2} - 3 \sqrt{2} y x - 3 \sqrt{2} x y + 9 \cdot (2^{\frac{2}{2}} x^{2}) = 1$

Step 3.7.4.2

Rewrite the expression.

$x^{2} + y^{2} - 3 \sqrt{2} y x - 3 \sqrt{2} x y + 9 \cdot (2 x^{2}) = 1$

Step 3.7.5

Evaluate the exponent.

$x^{2} + y^{2} - 3 \sqrt{2} y x - 3 \sqrt{2} x y + 9 \cdot (2 x^{2}) = 1$

Step 3.8

Multiply $9$ by $2$ .

$x^{2} + y^{2} - 3 \sqrt{2} y x - 3 \sqrt{2} x y + 18 x^{2} = 1$

Step 4

Reorder the factors in the terms $- 3 \sqrt{2} y x$ and $- 3 \sqrt{2} x y$ .

$x^{2} + y^{2} - 3 x y \sqrt{2} - 3 x y \sqrt{2} + 18 x^{2} = 1$

Step 5

Subtract $3 x y \sqrt{2}$ from $- 3 x y \sqrt{2}$ .

$x^{2} + y^{2} - 6 x y \sqrt{2} + 18 x^{2} = 1$

Step 6

Add $x^{2}$ and $18 x^{2}$ .

$19 x^{2} + y^{2} - 6 x y \sqrt{2} = 1$

Step 7

Subtract $1$ from both sides of the equation.

$19 x^{2} + y^{2} - 6 x y \sqrt{2} - 1 = 0$

Step 8

Use the quadratic formula to find the solutions.

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

Step 9

Substitute the values $a = 19$ , $b = - 6 y \sqrt{2}$ , and $c = y^{2} - 1$ into the quadratic formula and solve for $x$ .

$\frac{6 y \sqrt{2} \pm \sqrt{{(- 6 y \sqrt{2})}^{2} - 4 \cdot (19 \cdot (y^{2} - 1))}}{2 \cdot 19}$

Step 10

Simplify.

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Step 10.1

Simplify the numerator.

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Step 10.1.1

Add parentheses.

$x = \frac{6 y \sqrt{2} \pm \sqrt{{(- 6 (y \sqrt{2}))}^{2} - 4 \cdot (19 \cdot (y^{2} - 1))}}{2 \cdot 19}$

Step 10.1.2

Let $u = 19 \cdot (y^{2} - 1)$ . Substitute $u$ for all occurrences of $19 \cdot (y^{2} - 1)$ .

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Step 10.1.2.1

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

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Step 10.1.2.1.1

Apply the product rule to $- 6 y \sqrt{2}$ .

$x = \frac{6 y \sqrt{2} \pm \sqrt{{(- 6 y)}^{2} {\sqrt{2}}^{2} - 4 \cdot u}}{2 \cdot 19}$

Step 10.1.2.1.2

Apply the product rule to $- 6 y$ .

$x = \frac{6 y \sqrt{2} \pm \sqrt{{(- 6)}^{2} y^{2} {\sqrt{2}}^{2} - 4 \cdot u}}{2 \cdot 19}$

Step 10.1.2.2

Raise $- 6$ to the power of $2$ .

$x = \frac{6 y \sqrt{2} \pm \sqrt{36 y^{2} {\sqrt{2}}^{2} - 4 \cdot u}}{2 \cdot 19}$

Step 10.1.2.3

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 10.1.2.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$x = \frac{6 y \sqrt{2} \pm \sqrt{36 y^{2} {(2^{\frac{1}{2}})}^{2} - 4 \cdot u}}{2 \cdot 19}$

Step 10.1.2.3.2

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

$x = \frac{6 y \sqrt{2} \pm \sqrt{36 y^{2} \cdot 2^{\frac{1}{2} \cdot 2} - 4 \cdot u}}{2 \cdot 19}$

Step 10.1.2.3.3

Combine $\frac{1}{2}$ and $2$ .

$x = \frac{6 y \sqrt{2} \pm \sqrt{36 y^{2} \cdot 2^{\frac{2}{2}} - 4 \cdot u}}{2 \cdot 19}$

Step 10.1.2.3.4

Cancel the common factor of $2$ .

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Step 10.1.2.3.4.1

Cancel the common factor.

$x = \frac{6 y \sqrt{2} \pm \sqrt{36 y^{2} \cdot 2^{\frac{2}{2}} - 4 \cdot u}}{2 \cdot 19}$

Step 10.1.2.3.4.2

Rewrite the expression.

$x = \frac{6 y \sqrt{2} \pm \sqrt{36 y^{2} \cdot 2 - 4 \cdot u}}{2 \cdot 19}$

Step 10.1.2.3.5

Evaluate the exponent.

$x = \frac{6 y \sqrt{2} \pm \sqrt{36 y^{2} \cdot 2 - 4 \cdot u}}{2 \cdot 19}$

Step 10.1.2.4

Multiply $2$ by $36$ .

$x = \frac{6 y \sqrt{2} \pm \sqrt{72 y^{2} - 4 u}}{2 \cdot 19}$

Step 10.1.3

Factor $4$ out of $72 y^{2} - 4 u$ .

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Step 10.1.3.1

Factor $4$ out of $72 y^{2}$ .

$x = \frac{6 y \sqrt{2} \pm \sqrt{4 (18 y^{2}) - 4 u}}{2 \cdot 19}$

Step 10.1.3.2

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

$x = \frac{6 y \sqrt{2} \pm \sqrt{4 (18 y^{2}) + 4 (- u)}}{2 \cdot 19}$

Step 10.1.3.3

Factor $4$ out of $4 (18 y^{2}) + 4 (- u)$ .

$x = \frac{6 y \sqrt{2} \pm \sqrt{4 (18 y^{2} - u)}}{2 \cdot 19}$

Step 10.1.4

Replace all occurrences of $u$ with $19 \cdot (y^{2} - 1)$ .

$x = \frac{6 y \sqrt{2} \pm \sqrt{4 (18 y^{2} - (19 \cdot (y^{2} - 1)))}}{2 \cdot 19}$

Step 10.1.5

Simplify.

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Step 10.1.5.1

Simplify each term.

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Step 10.1.5.1.1

Apply the distributive property.

$x = \frac{6 y \sqrt{2} \pm \sqrt{4 (18 y^{2} - (19 y^{2} + 19 \cdot - 1))}}{2 \cdot 19}$

Step 10.1.5.1.2

Multiply $19$ by $- 1$ .

$x = \frac{6 y \sqrt{2} \pm \sqrt{4 (18 y^{2} - (19 y^{2} - 19))}}{2 \cdot 19}$

Step 10.1.5.1.3

Apply the distributive property.

$x = \frac{6 y \sqrt{2} \pm \sqrt{4 (18 y^{2} - (19 y^{2}) + 19)}}{2 \cdot 19}$

Step 10.1.5.1.4

Multiply $19$ by $- 1$ .

$x = \frac{6 y \sqrt{2} \pm \sqrt{4 (18 y^{2} - 19 y^{2} + 19)}}{2 \cdot 19}$

Step 10.1.5.1.5

Multiply $- 1$ by $- 19$ .

$x = \frac{6 y \sqrt{2} \pm \sqrt{4 (18 y^{2} - 19 y^{2} + 19)}}{2 \cdot 19}$

Step 10.1.5.2

Subtract $19 y^{2}$ from $18 y^{2}$ .

$x = \frac{6 y \sqrt{2} \pm \sqrt{4 (- y^{2} + 19)}}{2 \cdot 19}$

Step 10.1.6

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

$x = \frac{6 y \sqrt{2} \pm \sqrt{2^{2} (- y^{2} + 19)}}{2 \cdot 19}$

Step 10.1.7

Pull terms out from under the radical.

$x = \frac{6 y \sqrt{2} \pm 2 \sqrt{- y^{2} + 19}}{2 \cdot 19}$

Step 10.2

Multiply $2$ by $19$ .

$x = \frac{6 y \sqrt{2} \pm 2 \sqrt{- y^{2} + 19}}{38}$

Step 10.3

Simplify $\frac{6 y \sqrt{2} \pm 2 \sqrt{- y^{2} + 19}}{38}$ .

$x = \frac{3 y \sqrt{2} \pm \sqrt{- y^{2} + 19}}{19}$

Step 11

The final answer is the combination of both solutions.

$x = \frac{3 y \sqrt{2} + \sqrt{- y^{2} + 19}}{19}$

$x = \frac{3 y \sqrt{2} - \sqrt{- y^{2} + 19}}{19}$