Algebra Examples

$- 2 x^{2} - 10 x + 49$

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 $x = - \frac{b}{2 a}$ .

Tap for more steps...

Step 2.1

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

$x = - \frac{- 10}{2 (- 2)}$

Step 2.2

Remove parentheses.

$x = - \frac{- 10}{2 (- 2)}$

Step 2.3

Simplify $- \frac{- 10}{2 (- 2)}$ .

Tap for more steps...

Step 2.3.1

Cancel the common factor of $- 10$ and $2$ .

Tap for more steps...

Step 2.3.1.1

Factor $2$ out of $- 10$ .

$x = - \frac{2 \cdot - 5}{2 \cdot - 2}$

Step 2.3.1.2

Cancel the common factors.

Tap for more steps...

Step 2.3.1.2.1

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

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

Step 2.3.1.2.2

Cancel the common factor.

$x = - \frac{2 \cdot - 5}{2 \cdot - 2}$

Step 2.3.1.2.3

Rewrite the expression.

$x = - \frac{- 5}{- 2}$

Step 2.3.2

Dividing two negative values results in a positive value.

$x = - \frac{5}{2}$

Step 3

Evaluate $f (- \frac{5}{2})$ .

Tap for more steps...

Step 3.1

Replace the variable $x$ with $- \frac{5}{2}$ in the expression.

$f (- \frac{5}{2}) = - 2 {(- \frac{5}{2})}^{2} - 10 (- \frac{5}{2}) + 49$

Step 3.2

Simplify the result.

Tap for more steps...

Step 3.2.1

Simplify each term.

Tap for more steps...

Step 3.2.1.1

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

Tap for more steps...

Step 3.2.1.1.1

Apply the product rule to $- \frac{5}{2}$ .

$f (- \frac{5}{2}) = - 2 ({(- 1)}^{2} {(\frac{5}{2})}^{2}) - 10 (- \frac{5}{2}) + 49$

Step 3.2.1.1.2

Apply the product rule to $\frac{5}{2}$ .

$f (- \frac{5}{2}) = - 2 ({(- 1)}^{2} (\frac{5^{2}}{2^{2}})) - 10 (- \frac{5}{2}) + 49$

Step 3.2.1.2

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

$f (- \frac{5}{2}) = - 2 (1 (\frac{5^{2}}{2^{2}})) - 10 (- \frac{5}{2}) + 49$

Step 3.2.1.3

Multiply $\frac{5^{2}}{2^{2}}$ by $1$ .

$f (- \frac{5}{2}) = - 2 \frac{5^{2}}{2^{2}} - 10 (- \frac{5}{2}) + 49$

Step 3.2.1.4

Raise $5$ to the power of $2$ .

$f (- \frac{5}{2}) = - 2 \frac{25}{2^{2}} - 10 (- \frac{5}{2}) + 49$

Step 3.2.1.5

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

$f (- \frac{5}{2}) = - 2 (\frac{25}{4}) - 10 (- \frac{5}{2}) + 49$

Step 3.2.1.6

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.2.1.6.1

Factor $2$ out of $- 2$ .

$f (- \frac{5}{2}) = 2 (- 1) (\frac{25}{4}) - 10 (- \frac{5}{2}) + 49$

Step 3.2.1.6.2

Factor $2$ out of $4$ .

$f (- \frac{5}{2}) = 2 \cdot (- 1 \frac{25}{2 \cdot 2}) - 10 (- \frac{5}{2}) + 49$

Step 3.2.1.6.3

Cancel the common factor.

$f (- \frac{5}{2}) = 2 \cdot (- 1 \frac{25}{2 \cdot 2}) - 10 (- \frac{5}{2}) + 49$

Step 3.2.1.6.4

Rewrite the expression.

$f (- \frac{5}{2}) = - 1 (\frac{25}{2}) - 10 (- \frac{5}{2}) + 49$

Step 3.2.1.7

Rewrite $- 1 (\frac{25}{2})$ as $- (\frac{25}{2})$ .

$f (- \frac{5}{2}) = - (\frac{25}{2}) - 10 (- \frac{5}{2}) + 49$

Step 3.2.1.8

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.2.1.8.1

Move the leading negative in $- \frac{5}{2}$ into the numerator.

$f (- \frac{5}{2}) = - \frac{25}{2} - 10 (\frac{- 5}{2}) + 49$

Step 3.2.1.8.2

Factor $2$ out of $- 10$ .

$f (- \frac{5}{2}) = - \frac{25}{2} + 2 (- 5) (\frac{- 5}{2}) + 49$

Step 3.2.1.8.3

Cancel the common factor.

$f (- \frac{5}{2}) = - \frac{25}{2} + 2 \cdot (- 5 (\frac{- 5}{2})) + 49$

Step 3.2.1.8.4

Rewrite the expression.

$f (- \frac{5}{2}) = - \frac{25}{2} - 5 \cdot - 5 + 49$

Step 3.2.1.9

Multiply $- 5$ by $- 5$ .

$f (- \frac{5}{2}) = - \frac{25}{2} + 25 + 49$

Step 3.2.2

Find the common denominator.

Tap for more steps...

Step 3.2.2.1

Write $25$ as a fraction with denominator $1$ .

$f (- \frac{5}{2}) = - \frac{25}{2} + \frac{25}{1} + 49$

Step 3.2.2.2

Multiply $\frac{25}{1}$ by $\frac{2}{2}$ .

$f (- \frac{5}{2}) = - \frac{25}{2} + \frac{25}{1} \cdot \frac{2}{2} + 49$

Step 3.2.2.3

Multiply $\frac{25}{1}$ by $\frac{2}{2}$ .

$f (- \frac{5}{2}) = - \frac{25}{2} + \frac{25 \cdot 2}{2} + 49$

Step 3.2.2.4

Write $49$ as a fraction with denominator $1$ .

$f (- \frac{5}{2}) = - \frac{25}{2} + \frac{25 \cdot 2}{2} + \frac{49}{1}$

Step 3.2.2.5

Multiply $\frac{49}{1}$ by $\frac{2}{2}$ .

$f (- \frac{5}{2}) = - \frac{25}{2} + \frac{25 \cdot 2}{2} + \frac{49}{1} \cdot \frac{2}{2}$

Step 3.2.2.6

Multiply $\frac{49}{1}$ by $\frac{2}{2}$ .

$f (- \frac{5}{2}) = - \frac{25}{2} + \frac{25 \cdot 2}{2} + \frac{49 \cdot 2}{2}$

Step 3.2.3

Combine the numerators over the common denominator.

$f (- \frac{5}{2}) = \frac{- 25 + 25 \cdot 2 + 49 \cdot 2}{2}$

Step 3.2.4

Simplify each term.

Tap for more steps...

Step 3.2.4.1

Multiply $25$ by $2$ .

$f (- \frac{5}{2}) = \frac{- 25 + 50 + 49 \cdot 2}{2}$

Step 3.2.4.2

Multiply $49$ by $2$ .

$f (- \frac{5}{2}) = \frac{- 25 + 50 + 98}{2}$

Step 3.2.5

Simplify by adding numbers.

Tap for more steps...

Step 3.2.5.1

Add $- 25$ and $50$ .

$f (- \frac{5}{2}) = \frac{25 + 98}{2}$

Step 3.2.5.2

Add $25$ and $98$ .

$f (- \frac{5}{2}) = \frac{123}{2}$

Step 3.2.6

The final answer is $\frac{123}{2}$ .

$\frac{123}{2}$

Step 4

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

$(- \frac{5}{2}, \frac{123}{2})$

Step 5