Precalculus Examples

$f (x) = 3 x^{2} + 24 x - 8$

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

Tap for more steps...

Step 2.1

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

$x = - \frac{24}{2 (3)}$

Step 2.2

Remove parentheses.

$x = - \frac{24}{2 (3)}$

Step 2.3

Simplify $- \frac{24}{2 (3)}$ .

Tap for more steps...

Step 2.3.1

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

Tap for more steps...

Step 2.3.1.1

Factor $2$ out of $24$ .

$x = - \frac{2 \cdot 12}{2 \cdot 3}$

Step 2.3.1.2

Cancel the common factors.

Tap for more steps...

Step 2.3.1.2.1

Factor $2$ out of $2 \cdot 3$ .

$x = - \frac{2 \cdot 12}{2 (3)}$

Step 2.3.1.2.2

Cancel the common factor.

$x = - \frac{2 \cdot 12}{2 \cdot 3}$

Step 2.3.1.2.3

Rewrite the expression.

$x = - \frac{12}{3}$

Step 2.3.2

Cancel the common factor of $12$ and $3$ .

Tap for more steps...

Step 2.3.2.1

Factor $3$ out of $12$ .

$x = - \frac{3 \cdot 4}{3}$

Step 2.3.2.2

Cancel the common factors.

Tap for more steps...

Step 2.3.2.2.1

Factor $3$ out of $3$ .

$x = - \frac{3 \cdot 4}{3 (1)}$

Step 2.3.2.2.2

Cancel the common factor.

$x = - \frac{3 \cdot 4}{3 \cdot 1}$

Step 2.3.2.2.3

Rewrite the expression.

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

Step 2.3.2.2.4

Divide $4$ by $1$ .

$x = - 1 \cdot 4$

Step 2.3.3

Multiply $- 1$ by $4$ .

$x = - 4$

Step 3

Evaluate $f (- 4)$ .

Tap for more steps...

Step 3.1

Replace the variable $x$ with $- 4$ in the expression.

$f (- 4) = 3 {(- 4)}^{2} + 24 (- 4) - 8$

Step 3.2

Simplify the result.

Tap for more steps...

Step 3.2.1

Simplify each term.

Tap for more steps...

Step 3.2.1.1

Raise $- 4$ to the power of $2$ .

$f (- 4) = 3 \cdot 16 + 24 (- 4) - 8$

Step 3.2.1.2

Multiply $3$ by $16$ .

$f (- 4) = 48 + 24 (- 4) - 8$

Step 3.2.1.3

Multiply $24$ by $- 4$ .

$f (- 4) = 48 - 96 - 8$

Step 3.2.2

Simplify by subtracting numbers.

Tap for more steps...

Step 3.2.2.1

Subtract $96$ from $48$ .

$f (- 4) = - 48 - 8$

Step 3.2.2.2

Subtract $8$ from $- 48$ .

$f (- 4) = - 56$

Step 3.2.3

The final answer is $- 56$ .

$- 56$

Step 4

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

$(- 4, - 56)$

Step 5