Find the Maximum/Minimum Value 4x^2-12x

$4 x^{2} - 12 x$

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

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

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

$x = - \frac{- 12}{2 (4)}$

Remove parentheses.

$x = - \frac{- 12}{2 (4)}$

Simplify $- \frac{- 12}{2 (4)}$ .

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

Factor $2$ out of $- 12$ .

$x = - \frac{2 \cdot - 6}{2 \cdot 4}$

Cancel the common factors.

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

$x = - \frac{2 \cdot - 6}{2 (4)}$

Cancel the common factor.

$x = - \frac{2 \cdot - 6}{2 \cdot 4}$

Rewrite the expression.

$x = - \frac{- 6}{4}$

Cancel the common factor of $- 6$ and $4$ .

Factor $2$ out of $- 6$ .

$x = - \frac{2 (- 3)}{4}$

Cancel the common factors.

Factor $2$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Move the negative in front of the fraction.

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

Multiply $- - \frac{3}{2}$ .

Multiply $- 1$ by $- 1$ .

$x = 1 (\frac{3}{2})$

Multiply $\frac{3}{2}$ by $1$ .

$x = \frac{3}{2}$

Evaluate $f (\frac{3}{2})$ .

Replace the variable $x$ with $\frac{3}{2}$ in the expression.

$f (\frac{3}{2}) = 4 {(\frac{3}{2})}^{2} - 12 (\frac{3}{2})$

Simplify the result.

Simplify each term.

Apply the product rule to $\frac{3}{2}$ .

$f (\frac{3}{2}) = 4 (\frac{3^{2}}{2^{2}}) - 12 (\frac{3}{2})$

Raise $3$ to the power of $2$ .

$f (\frac{3}{2}) = 4 (\frac{9}{2^{2}}) - 12 (\frac{3}{2})$

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

$f (\frac{3}{2}) = 4 (\frac{9}{4}) - 12 (\frac{3}{2})$

Cancel the common factor of $4$ .

Cancel the common factor.

$f (\frac{3}{2}) = 4 (\frac{9}{4}) - 12 (\frac{3}{2})$

Rewrite the expression.

$f (\frac{3}{2}) = 9 - 12 (\frac{3}{2})$

Cancel the common factor of $2$ .

Factor $2$ out of $- 12$ .

$f (\frac{3}{2}) = 9 + 2 (- 6) (\frac{3}{2})$

Cancel the common factor.

$f (\frac{3}{2}) = 9 + 2 \cdot (- 6 (\frac{3}{2}))$

Rewrite the expression.

$f (\frac{3}{2}) = 9 - 6 \cdot 3$

Multiply $- 6$ by $3$ .

$f (\frac{3}{2}) = 9 - 18$

Subtract $18$ from $9$ .

$f (\frac{3}{2}) = - 9$

The final answer is $- 9$ .

$- 9$

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

$(\frac{3}{2}, - 9)$

Find the Maximum/Minimum Value 4x^2-12x

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