Find the Maximum/Minimum Value f(x)=-5x^2-50x-123

$f (x) = - 5 x^{2} - 50 x - 123$

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

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

$x = - \frac{- 50}{2 (- 5)}$

Remove parentheses.

$x = - \frac{- 50}{2 (- 5)}$

Simplify $- \frac{- 50}{2 (- 5)}$ .

Cancel the common factor of $- 50$ and $2$ .

Factor $2$ out of $- 50$ .

$x = - \frac{2 \cdot - 25}{2 \cdot - 5}$

Cancel the common factors.

Factor $2$ out of $2 \cdot - 5$ .

$x = - \frac{2 \cdot - 25}{2 (- 5)}$

Cancel the common factor.

$x = - \frac{2 \cdot - 25}{2 \cdot - 5}$

Rewrite the expression.

$x = - \frac{- 25}{- 5}$

Cancel the common factor of $- 25$ and $- 5$ .

Factor $- 5$ out of $- 25$ .

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

Cancel the common factors.

Factor $- 5$ out of $- 5$ .

$x = - \frac{- 5 \cdot 5}{- 5 \cdot 1}$

Cancel the common factor.

$x = - \frac{- 5 \cdot 5}{- 5 \cdot 1}$

Rewrite the expression.

$x = - \frac{5}{1}$

Divide $5$ by $1$ .

$x = - 1 \cdot 5$

Multiply $- 1$ by $5$ .

$x = - 5$

Step 3

Evaluate $f (- 5)$ .

Replace the variable $x$ with $- 5$ in the expression.

$f (- 5) = - 5 {(- 5)}^{2} - 50 \cdot - 5 - 123$

Simplify the result.

Simplify each term.

Multiply $- 5$ by ${(- 5)}^{2}$ by adding the exponents.

Multiply $- 5$ by ${(- 5)}^{2}$ .

Raise $- 5$ to the power of $1$ .

$f (- 5) = (- 5) {(- 5)}^{2} - 50 \cdot - 5 - 123$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (- 5) = {(- 5)}^{1 + 2} - 50 \cdot - 5 - 123$

Add $1$ and $2$ .

$f (- 5) = {(- 5)}^{3} - 50 \cdot - 5 - 123$

Raise $- 5$ to the power of $3$ .

$f (- 5) = - 125 - 50 \cdot - 5 - 123$

Multiply $- 50$ by $- 5$ .

$f (- 5) = - 125 + 250 - 123$

Simplify by adding and subtracting.

Add $- 125$ and $250$ .

$f (- 5) = 125 - 123$

Subtract $123$ from $125$ .

$f (- 5) = 2$

The final answer is $2$ .

$2$

Step 4

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

$(- 5, 2)$

Step 5

Find the Maximum/Minimum Value f(x)=-5x^2-50x-123

Related Questions

Support LCMGCF.com by disabling your adblocker.