Algebra Examples

$f (x) = a {(x - h)}^{2} + k$

Step 1

Simplify each term.

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Step 1.1

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

$f (x) = a ((x - h) (x - h)) + k$

Step 1.2

Expand $(x - h) (x - h)$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

$f (x) = a (x (x - h) - h (x - h)) + k$

Step 1.2.2

Apply the distributive property.

$f (x) = a (x \cdot x + x (- h) - h (x - h)) + k$

Step 1.2.3

Apply the distributive property.

$f (x) = a (x \cdot x + x (- h) - h x - h (- h)) + k$

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Multiply $x$ by $x$ .

$f (x) = a (x^{2} + x (- h) - h x - h (- h)) + k$

Step 1.3.1.2

Rewrite using the commutative property of multiplication.

$f (x) = a (x^{2} - x h - h x - h (- h)) + k$

Step 1.3.1.3

Rewrite using the commutative property of multiplication.

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

Step 1.3.1.4

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

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Step 1.3.1.4.1

Move $h$ .

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

Step 1.3.1.4.2

Multiply $h$ by $h$ .

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

Step 1.3.1.5

Multiply $- 1$ by $- 1$ .

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

Step 1.3.1.6

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

$f (x) = a (x^{2} - x h - h x + h^{2}) + k$

Step 1.3.2

Subtract $h x$ from $- x h$ .

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Step 1.3.2.1

Move $x$ .

$f (x) = a (x^{2} - h x - h x + h^{2}) + k$

Step 1.3.2.2

Subtract $h x$ from $- h x$ .

$f (x) = a (x^{2} - 2 h x + h^{2}) + k$

Step 1.4

Apply the distributive property.

$f (x) = a x^{2} + a (- 2 h x) + a h^{2} + k$

Step 1.5

Rewrite using the commutative property of multiplication.

$f (x) = a x^{2} - 2 a h x + a h^{2} + k$

Step 2

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$a x^{2} - 2 a h x + a h^{2} + k = f (x)$

Step 3

Subtract $f (x)$ from both sides of the equation.

$a x^{2} - 2 a h x + a h^{2} + k - f (x) = 0$

Step 4

Use the quadratic formula to find the solutions.

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

Step 5

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

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

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Add parentheses.

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

Step 6.1.2

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

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Step 6.1.2.1

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

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Step 6.1.2.1.1

Apply the product rule to $- 2 a h$ .

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

Step 6.1.2.1.2

Apply the product rule to $- 2 a$ .

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

Step 6.1.2.2

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

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

Step 6.1.3

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

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Step 6.1.3.1

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

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

Step 6.1.3.2

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

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

Step 6.1.3.3

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

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

Step 6.1.4

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

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

Step 6.1.5

Simplify.

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Step 6.1.5.1

Simplify each term.

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Step 6.1.5.1.1

Apply the distributive property.

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

Step 6.1.5.1.2

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

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Step 6.1.5.1.2.1

Move $a$ .

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

Step 6.1.5.1.2.2

Multiply $a$ by $a$ .

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

Step 6.1.5.1.3

Apply the distributive property.

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

Step 6.1.5.1.4

Multiply $- (a (- f (x)))$ .

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Step 6.1.5.1.4.1

Multiply $- 1$ by $- 1$ .

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

Step 6.1.5.1.4.2

Multiply $a$ by $1$ .

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

Step 6.1.5.2

Subtract $a^{2} h^{2}$ from $a^{2} h^{2}$ .

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

Step 6.1.5.3

Subtract $a k$ from $0$ .

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

Step 6.1.6

Factor $a$ out of $- a k + a f (x)$ .

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Step 6.1.6.1

Factor $a$ out of $- a k$ .

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

Step 6.1.6.2

Factor $a$ out of $a f (x)$ .

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

Step 6.1.6.3

Factor $a$ out of $a (- 1 k) + a (f (x))$ .

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

Step 6.1.7

Rewrite $- 1 k$ as $- k$ .

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

Step 6.1.8

Rewrite $4 a (- k + f (x))$ as $2^{2} (a (- k + f (x)))$ .

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Step 6.1.8.1

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

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

Step 6.1.8.2

Add parentheses.

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

Step 6.1.9

Pull terms out from under the radical.

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

Step 6.2

Simplify $\frac{2 a h \pm 2 \sqrt{a (- k + f (x))}}{2 a}$ .

$x = \frac{a h \pm \sqrt{a (- k + f (x))}}{a}$

Step 7

The final answer is the combination of both solutions.

$x = \frac{a h + \sqrt{a (- k + f (x))}}{a}, \frac{a h - \sqrt{a (- k + f (x))}}{a}$