Find the Maximum/Minimum Value y=4x^2+40x+44

$y = 4 x^{2} + 40 x + 44$

Step 1

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

$f_{min}$ $x = a x^{2} + b x + c$ 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 $b$ .

$x = - \frac{40}{2 (4)}$

Remove parentheses.

$x = - \frac{40}{2 (4)}$

Simplify $- \frac{40}{2 (4)}$ .

Cancel the common factor of $40$ and $2$ .

Factor $2$ out of $40$ .

$x = - \frac{2 \cdot 20}{2 \cdot 4}$

Cancel the common factors.

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

$x = - \frac{2 \cdot 20}{2 (4)}$

Cancel the common factor.

$x = - \frac{2 \cdot 20}{2 \cdot 4}$

Rewrite the expression.

$x = - \frac{20}{4}$

Cancel the common factor of $20$ and $4$ .

Factor $4$ out of $20$ .

$x = - \frac{4 \cdot 5}{4}$

Cancel the common factors.

Factor $4$ out of $4$ .

$x = - \frac{4 \cdot 5}{4 (1)}$

Cancel the common factor.

$x = - \frac{4 \cdot 5}{4 \cdot 1}$

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.

$f (- 5) = 4 {(- 5)}^{2} + 40 (- 5) + 44$

Simplify the result.

Simplify each term.

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

$f (- 5) = 4 \cdot 25 + 40 (- 5) + 44$

Multiply $4$ by $25$ .

$f (- 5) = 100 + 40 (- 5) + 44$

Multiply $40$ by $- 5$ .

$f (- 5) = 100 - 200 + 44$

Simplify by adding and subtracting.

Subtract $200$ from $100$ .

$f (- 5) = - 100 + 44$

Add $- 100$ and $44$ .

$f (- 5) = - 56$

The final answer is $- 56$ .

$- 56$

Step 4

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

$(- 5, - 56)$

Step 5

Find the Maximum/Minimum Value y=4x^2+40x+44

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