Find the Maximum/Minimum Value 2x^2+8x+3

$2 x^{2} + 8 x + 3$

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 $b$ .

$x = - \frac{8}{2 (2)}$

Remove parentheses.

$x = - \frac{8}{2 (2)}$

Simplify $- \frac{8}{2 (2)}$ .

Cancel the common factor of $8$ and $2$ .

Factor $2$ out of $8$ .

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $4$ and $2$ .

Factor $2$ out of $4$ .

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

Cancel the common factors.

Factor $2$ out of $2$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $2$ by $1$ .

$x = - 1 \cdot 2$

Multiply $- 1$ by $2$ .

$x = - 2$

Step 3

Evaluate $f (- 2)$ .

Replace the variable $x$ with $- 2$ in the expression.

$f (- 2) = 2 {(- 2)}^{2} + 8 (- 2) + 3$

Simplify the result.

Simplify each term.

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

$f (- 2) = 2 \cdot 4 + 8 (- 2) + 3$

Multiply $2$ by $4$ .

$f (- 2) = 8 + 8 (- 2) + 3$

Multiply $8$ by $- 2$ .

$f (- 2) = 8 - 16 + 3$

Simplify by adding and subtracting.

Subtract $16$ from $8$ .

$f (- 2) = - 8 + 3$

Add $- 8$ and $3$ .

$f (- 2) = - 5$

The final answer is $- 5$ .

$- 5$

Step 4

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

$(- 2, - 5)$

Step 5

Find the Maximum/Minimum Value 2x^2+8x+3

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