Algebra Examples

$f (x) = - 4 x^{2} - 12 x + 5$

Step 1

The maximum of a quadratic function occurs at $x = - \frac{b}{2 a}$ . If $a$ is negative, the maximum value of the function is $f (- \frac{b}{2 a})$ .

$f_{max}$ $x = a x^{2} + b x + c$ occurs at $x = - \frac{b}{2 a}$

Step 2

Find the value of $x = - \frac{b}{2 a}$ .

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Step 2.1

Substitute in the values of $a$ and $b$ .

$x = - \frac{- 12}{2 (- 4)}$

Step 2.2

Remove parentheses.

$x = - \frac{- 12}{2 (- 4)}$

Step 2.3

Simplify $- \frac{- 12}{2 (- 4)}$ .

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Step 2.3.1

Cancel the common factor of $- 12$ and $2$ .

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Step 2.3.1.1

Factor $2$ out of $- 12$ .

$x = - \frac{2 \cdot - 6}{2 \cdot - 4}$

Step 2.3.1.2

Cancel the common factors.

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Step 2.3.1.2.1

Factor $2$ out of $2 \cdot - 4$ .

$x = - \frac{2 \cdot - 6}{2 (- 4)}$

Step 2.3.1.2.2

Cancel the common factor.

$x = - \frac{2 \cdot - 6}{2 \cdot - 4}$

Step 2.3.1.2.3

Rewrite the expression.

$x = - \frac{- 6}{- 4}$

Step 2.3.2

Cancel the common factor of $- 6$ and $- 4$ .

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Step 2.3.2.1

Factor $- 2$ out of $- 6$ .

$x = - \frac{- 2 (3)}{- 4}$

Step 2.3.2.2

Cancel the common factors.

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Step 2.3.2.2.1

Factor $- 2$ out of $- 4$ .

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

Step 2.3.2.2.2

Cancel the common factor.

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

Step 2.3.2.2.3

Rewrite the expression.

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

Step 3

Evaluate $f (- \frac{3}{2})$ .

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Step 3.1

Replace the variable $x$ with $- \frac{3}{2}$ in the expression.

$f (- \frac{3}{2}) = - 4 {(- \frac{3}{2})}^{2} - 12 (- \frac{3}{2}) + 5$

Step 3.2

Simplify the result.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

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

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Step 3.2.1.1.1

Apply the product rule to $- \frac{3}{2}$ .

$f (- \frac{3}{2}) = - 4 ({(- 1)}^{2} {(\frac{3}{2})}^{2}) - 12 (- \frac{3}{2}) + 5$

Step 3.2.1.1.2

Apply the product rule to $\frac{3}{2}$ .

$f (- \frac{3}{2}) = - 4 ({(- 1)}^{2} (\frac{3^{2}}{2^{2}})) - 12 (- \frac{3}{2}) + 5$

Step 3.2.1.2

Raise $- 1$ to the power of $2$ .

$f (- \frac{3}{2}) = - 4 (1 (\frac{3^{2}}{2^{2}})) - 12 (- \frac{3}{2}) + 5$

Step 3.2.1.3

Multiply $\frac{3^{2}}{2^{2}}$ by $1$ .

$f (- \frac{3}{2}) = - 4 \frac{3^{2}}{2^{2}} - 12 (- \frac{3}{2}) + 5$

Step 3.2.1.4

Raise $3$ to the power of $2$ .

$f (- \frac{3}{2}) = - 4 \frac{9}{2^{2}} - 12 (- \frac{3}{2}) + 5$

Step 3.2.1.5

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

$f (- \frac{3}{2}) = - 4 (\frac{9}{4}) - 12 (- \frac{3}{2}) + 5$

Step 3.2.1.6

Cancel the common factor of $4$ .

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Step 3.2.1.6.1

Factor $4$ out of $- 4$ .

$f (- \frac{3}{2}) = 4 (- 1) (\frac{9}{4}) - 12 (- \frac{3}{2}) + 5$

Step 3.2.1.6.2

Cancel the common factor.

$f (- \frac{3}{2}) = 4 \cdot (- 1 (\frac{9}{4})) - 12 (- \frac{3}{2}) + 5$

Step 3.2.1.6.3

Rewrite the expression.

$f (- \frac{3}{2}) = - 1 \cdot 9 - 12 (- \frac{3}{2}) + 5$

Step 3.2.1.7

Multiply $- 1$ by $9$ .

$f (- \frac{3}{2}) = - 9 - 12 (- \frac{3}{2}) + 5$

Step 3.2.1.8

Cancel the common factor of $2$ .

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Step 3.2.1.8.1

Move the leading negative in $- \frac{3}{2}$ into the numerator.

$f (- \frac{3}{2}) = - 9 - 12 (\frac{- 3}{2}) + 5$

Step 3.2.1.8.2

Factor $2$ out of $- 12$ .

$f (- \frac{3}{2}) = - 9 + 2 (- 6) (\frac{- 3}{2}) + 5$

Step 3.2.1.8.3

Cancel the common factor.

$f (- \frac{3}{2}) = - 9 + 2 \cdot (- 6 (\frac{- 3}{2})) + 5$

Step 3.2.1.8.4

Rewrite the expression.

$f (- \frac{3}{2}) = - 9 - 6 \cdot - 3 + 5$

Step 3.2.1.9

Multiply $- 6$ by $- 3$ .

$f (- \frac{3}{2}) = - 9 + 18 + 5$

Step 3.2.2

Simplify by adding numbers.

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Step 3.2.2.1

Add $- 9$ and $18$ .

$f (- \frac{3}{2}) = 9 + 5$

Step 3.2.2.2

Add $9$ and $5$ .

$f (- \frac{3}{2}) = 14$

Step 3.2.3

The final answer is $14$ .

$14$

Step 4

Use the $x$ and $y$ values to find where the maximum occurs.

$(- \frac{3}{2}, 14)$

Step 5