Algebra Examples

$f (x) = - 5 x^{2} - 60 x - 181$

Step 1

Write $f (x) = - 5 x^{2} - 60 x - 181$ as an equation.

$y = - 5 x^{2} - 60 x - 181$

Step 2

Complete the square for $- 5 x^{2} - 60 x - 181$ .

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Step 2.1

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = - 5$

$b = - 60$

$c = - 181$

Step 2.2

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Step 2.3

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

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Step 2.3.1

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{- 60}{2 \cdot - 5}$

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

Cancel the common factor of $- 60$ and $2$ .

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Step 2.3.2.1.1

Factor $2$ out of $- 60$ .

$d = \frac{2 \cdot - 30}{2 \cdot - 5}$

Step 2.3.2.1.2

Cancel the common factors.

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Step 2.3.2.1.2.1

Factor $2$ out of $2 \cdot - 5$ .

$d = \frac{2 \cdot - 30}{2 (- 5)}$

Step 2.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot - 30}{2 \cdot - 5}$

Step 2.3.2.1.2.3

Rewrite the expression.

$d = \frac{- 30}{- 5}$

Step 2.3.2.2

Cancel the common factor of $- 30$ and $- 5$ .

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Step 2.3.2.2.1

Factor $- 5$ out of $- 30$ .

$d = \frac{- 5 (6)}{- 5}$

Step 2.3.2.2.2

Cancel the common factors.

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Step 2.3.2.2.2.1

Factor $- 5$ out of $- 5$ .

$d = \frac{- 5 \cdot 6}{- 5 \cdot 1}$

Step 2.3.2.2.2.2

Cancel the common factor.

$d = \frac{- 5 \cdot 6}{- 5 \cdot 1}$

Step 2.3.2.2.2.3

Rewrite the expression.

$d = \frac{6}{1}$

Step 2.3.2.2.2.4

Divide $6$ by $1$ .

$d = 6$

Step 2.4

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

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Step 2.4.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = - 181 - \frac{{(- 60)}^{2}}{4 \cdot - 5}$

Step 2.4.2

Simplify the right side.

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Step 2.4.2.1

Simplify each term.

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Step 2.4.2.1.1

Raise $- 60$ to the power of $2$ .

$e = - 181 - \frac{3600}{4 \cdot - 5}$

Step 2.4.2.1.2

Multiply $4$ by $- 5$ .

$e = - 181 - \frac{3600}{- 20}$

Step 2.4.2.1.3

Divide $3600$ by $- 20$ .

$e = - 181 - - 180$

Step 2.4.2.1.4

Multiply $- 1$ by $- 180$ .

$e = - 181 + 180$

Step 2.4.2.2

Add $- 181$ and $180$ .

$e = - 1$

Step 2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 5 {(x + 6)}^{2} - 1$ .

$- 5 {(x + 6)}^{2} - 1$

Step 3

Set $y$ equal to the new right side.

$y = - 5 {(x + 6)}^{2} - 1$