Find the Maximum/Minimum Value y=2x^2+24x+79

$y = 2 x^{2} + 24 x + 79$

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 $b$ .

$x = - \frac{24}{2 (2)}$

Remove parentheses.

$x = - \frac{24}{2 (2)}$

Simplify $- \frac{24}{2 (2)}$ .

Cancel the common factor of $24$ and $2$ .

Factor $2$ out of $24$ .

$x = - \frac{2 \cdot 12}{2 \cdot 2}$

Cancel the common factors.

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

$x = - \frac{2 \cdot 12}{2 (2)}$

Cancel the common factor.

$x = - \frac{2 \cdot 12}{2 \cdot 2}$

Rewrite the expression.

$x = - \frac{12}{2}$

Cancel the common factor of $12$ and $2$ .

Factor $2$ out of $12$ .

$x = - \frac{2 \cdot 6}{2}$

Cancel the common factors.

Factor $2$ out of $2$ .

$x = - \frac{2 \cdot 6}{2 (1)}$

Cancel the common factor.

$x = - \frac{2 \cdot 6}{2 \cdot 1}$

Rewrite the expression.

$x = - \frac{6}{1}$

Divide $6$ by $1$ .

$x = - 1 \cdot 6$

Multiply $- 1$ by $6$ .

$x = - 6$

Step 3

Evaluate $f (- 6)$ .

Replace the variable $x$ with $- 6$ in the expression.

$f (- 6) = 2 {(- 6)}^{2} + 24 (- 6) + 79$

Simplify the result.

Simplify each term.

Raise $- 6$ to the power of $2$ .

$f (- 6) = 2 \cdot 36 + 24 (- 6) + 79$

Multiply $2$ by $36$ .

$f (- 6) = 72 + 24 (- 6) + 79$

Multiply $24$ by $- 6$ .

$f (- 6) = 72 - 144 + 79$

Simplify by adding and subtracting.

Subtract $144$ from $72$ .

$f (- 6) = - 72 + 79$

Add $- 72$ and $79$ .

$f (- 6) = 7$

The final answer is $7$ .

$7$

Step 4

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

$(- 6, 7)$

Step 5

Find the Maximum/Minimum Value y=2x^2+24x+79

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