Algebra Examples

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

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

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

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 $- 7$ .

$x = 7$

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 $7$ .

$m (7) = - 49 + 98$

Step 3.2.2

Add $- 49$ and $98$ .

$m (7) = 49$

Step 3.2.3

The final answer is $49$ .

$49$

Step 4

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

$(7, 49)$

Step 5