Algebra Examples

$f (x) = - 2 x^{2} - 6 x + 9$

Step 1

Write $f (x) = - 2 x^{2} - 6 x + 9$ as an equation.

$y = - 2 x^{2} - 6 x + 9$

Step 2

Complete the square for $- 2 x^{2} - 6 x + 9$ .

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Step 2.1

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

$a = - 2$

$b = - 6$

$c = 9$

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 $d = \frac{b}{2 a}$ .

$d = \frac{- 6}{2 \cdot - 2}$

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

Cancel the common factor of $- 6$ and $2$ .

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Step 2.3.2.1.1

Factor $2$ out of $- 6$ .

$d = \frac{2 \cdot - 3}{2 \cdot - 2}$

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 $2 \cdot - 2$ .

$d = \frac{2 \cdot - 3}{2 (- 2)}$

Step 2.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot - 3}{2 \cdot - 2}$

Step 2.3.2.1.2.3

Rewrite the expression.

$d = \frac{- 3}{- 2}$

Step 2.3.2.2

Dividing two negative values results in a positive value.

$d = \frac{3}{2}$

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 $e = c - \frac{b^{2}}{4 a}$ .

$e = 9 - \frac{{(- 6)}^{2}}{4 \cdot - 2}$

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

$e = 9 - \frac{36}{4 \cdot - 2}$

Step 2.4.2.1.2

Multiply $4$ by $- 2$ .

$e = 9 - \frac{36}{- 8}$

Step 2.4.2.1.3

Cancel the common factor of $36$ and $- 8$ .

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Step 2.4.2.1.3.1

Factor $4$ out of $36$ .

$e = 9 - \frac{4 (9)}{- 8}$

Step 2.4.2.1.3.2

Cancel the common factors.

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Step 2.4.2.1.3.2.1

Factor $4$ out of $- 8$ .

$e = 9 - \frac{4 \cdot 9}{4 \cdot - 2}$

Step 2.4.2.1.3.2.2

Cancel the common factor.

$e = 9 - \frac{4 \cdot 9}{4 \cdot - 2}$

Step 2.4.2.1.3.2.3

Rewrite the expression.

$e = 9 - \frac{9}{- 2}$

Step 2.4.2.1.4

Move the negative in front of the fraction.

$e = 9 - - \frac{9}{2}$

Step 2.4.2.1.5

Multiply $- - \frac{9}{2}$ .

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Step 2.4.2.1.5.1

Multiply $- 1$ by $- 1$ .

$e = 9 + 1 (\frac{9}{2})$

Step 2.4.2.1.5.2

Multiply $\frac{9}{2}$ by $1$ .

$e = 9 + \frac{9}{2}$

Step 2.4.2.2

To write $9$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$e = 9 \cdot \frac{2}{2} + \frac{9}{2}$

Step 2.4.2.3

Combine $9$ and $\frac{2}{2}$ .

$e = \frac{9 \cdot 2}{2} + \frac{9}{2}$

Step 2.4.2.4

Combine the numerators over the common denominator.

$e = \frac{9 \cdot 2 + 9}{2}$

Step 2.4.2.5

Simplify the numerator.

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Step 2.4.2.5.1

Multiply $9$ by $2$ .

$e = \frac{18 + 9}{2}$

Step 2.4.2.5.2

Add $18$ and $9$ .

$e = \frac{27}{2}$

Step 2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 2 {(x + \frac{3}{2})}^{2} + \frac{27}{2}$ .

$- 2 {(x + \frac{3}{2})}^{2} + \frac{27}{2}$

Step 3

Set $y$ equal to the new right side.

$y = - 2 {(x + \frac{3}{2})}^{2} + \frac{27}{2}$