Algebra Examples

m (x) = - x^{2} + 14 x

$m (x) = - x^{2} + 14 x$

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

$b$ .

$x = - \frac{14}{2 (- 1)}$

Step 2.2

Remove parentheses.

$x = - \frac{14}{2 (- 1)}$

Step 2.3

Simplify

$- \frac{14}{2 (- 1)}$ .

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Step 2.3.1

Cancel the common factor of

$14$ and

$2$ .

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Step 2.3.1.1

Factor

$2$ out of

$14$ .

$x = - \frac{2 \cdot 7}{2 \cdot - 1}$

Step 2.3.1.2

Move the negative one from the denominator of

$\frac{7}{- 1}$ .

$x = - (- 1 \cdot 7)$

$x = - (- 1 \cdot 7)$

Step 2.3.2

Multiply

$- (- 1 \cdot 7)$ .

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Step 2.3.2.1

Multiply

$- 1$ by

$7$ .

$x = - - 7$

Step 2.3.2.2

Multiply

$- 1$ by

$- 7$ .

$x = 7$

$x = 7$

$x = 7$

$x = 7$

Step 3

Evaluate

$f (7)$ .

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Step 3.1

Replace the variable

$x$ with

$7$ in the expression.

$m (7) = - {(7)}^{2} + 14 (7)$

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

Raise

$7$ to the power of

$2$ .

$m (7) = - 1 \cdot 49 + 14 (7)$

Step 3.2.1.2

Multiply

$- 1$ by

$49$ .

$m (7) = - 49 + 14 (7)$

Step 3.2.1.3

Multiply

$14$ by

$7$ .

$m (7) = - 49 + 98$

$m (7) = - 49 + 98$

Step 3.2.2

Add

$- 49$ and

$98$ .

$m (7) = 49$

Step 3.2.3

The final answer is

$49$ .

$49$

$49$

$49$

Step 4

Use the

$x$ and

$y$ values to find where the maximum occurs.

$(7, 49)$

Step 5

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$