Find the Maximum/Minimum Value (x+12)^2-16

${(x + 12)}^{2} - 16$

Step 1

Simplify each term.

Rewrite ${(x + 12)}^{2}$ as $(x + 12) (x + 12)$ .

$f (x) = (x + 12) (x + 12) - 16$

Expand $(x + 12) (x + 12)$ using the FOIL Method.

Apply the distributive property.

$f (x) = x (x + 12) + 12 (x + 12) - 16$

Apply the distributive property.

$f (x) = x \cdot x + x \cdot 12 + 12 (x + 12) - 16$

Apply the distributive property.

$f (x) = x \cdot x + x \cdot 12 + 12 x + 12 \cdot 12 - 16$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$f (x) = x^{2} + x \cdot 12 + 12 x + 12 \cdot 12 - 16$

Move $12$ to the left of $x$ .

$f (x) = x^{2} + 12 \cdot x + 12 x + 12 \cdot 12 - 16$

Multiply $12$ by $12$ .

$f (x) = x^{2} + 12 x + 12 x + 144 - 16$

Add $12 x$ and $12 x$ .

$f (x) = x^{2} + 24 x + 144 - 16$

Step 2

Subtract $16$ from $144$ .

$f (x) = x^{2} + 24 x + 128$

Step 3

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 4

Find the value of $x = - \frac{b}{2 a}$ .

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

$x = - \frac{24}{2 (1)}$

Remove parentheses.

$x = - \frac{24}{2 (1)}$

Simplify $- \frac{24}{2 (1)}$ .

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

Factor $2$ out of $24$ .

$x = - \frac{2 \cdot 12}{2 \cdot 1}$

Cancel the common factors.

Factor $2$ out of $2 \cdot 1$ .

$x = - \frac{2 \cdot 12}{2 (1)}$

Cancel the common factor.

$x = - \frac{2 \cdot 12}{2 \cdot 1}$

Rewrite the expression.

$x = - \frac{12}{1}$

Divide $12$ by $1$ .

$x = - 1 \cdot 12$

Multiply $- 1$ by $12$ .

$x = - 12$

Step 5

Evaluate $f (- 12)$ .

Replace the variable $x$ with $- 12$ in the expression.

$f (- 12) = {(- 12)}^{2} + 24 (- 12) + 128$

Simplify the result.

Simplify each term.

Raise $- 12$ to the power of $2$ .

$f (- 12) = 144 + 24 (- 12) + 128$

Multiply $24$ by $- 12$ .

$f (- 12) = 144 - 288 + 128$

Simplify by adding and subtracting.

Subtract $288$ from $144$ .

$f (- 12) = - 144 + 128$

Add $- 144$ and $128$ .

$f (- 12) = - 16$

The final answer is $- 16$ .

$- 16$

Step 6

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

$(- 12, - 16)$

Step 7

Find the Maximum/Minimum Value (x+12)^2-16

Related Questions

Support LCMGCF.com by disabling your adblocker.