Find the Maximum/Minimum Value y=2(x-3)(x+5)

$y = 2 (x - 3) (x + 5)$

Step 1

Simplify by multiplying through.

Apply the distributive property.

$f (x) = (2 x + 2 \cdot - 3) (x + 5)$

Multiply $2$ by $- 3$ .

$f (x) = (2 x - 6) (x + 5)$

Step 2

Expand $(2 x - 6) (x + 5)$ using the FOIL Method.

Apply the distributive property.

$f (x) = 2 x (x + 5) - 6 (x + 5)$

Apply the distributive property.

$f (x) = 2 x \cdot x + 2 x \cdot 5 - 6 (x + 5)$

Apply the distributive property.

$f (x) = 2 x \cdot x + 2 x \cdot 5 - 6 x - 6 \cdot 5$

Step 3

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$f (x) = 2 (x \cdot x) + 2 x \cdot 5 - 6 x - 6 \cdot 5$

Multiply $x$ by $x$ .

$f (x) = 2 x^{2} + 2 x \cdot 5 - 6 x - 6 \cdot 5$

Multiply $5$ by $2$ .

$f (x) = 2 x^{2} + 10 x - 6 x - 6 \cdot 5$

Multiply $- 6$ by $5$ .

$f (x) = 2 x^{2} + 10 x - 6 x - 30$

Subtract $6 x$ from $10 x$ .

$f (x) = 2 x^{2} + 4 x - 30$

Step 4

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 5

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

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

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

Remove parentheses.

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

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

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

Factor $2$ out of $4$ .

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$x = - 1 \cdot 1$

Multiply $- 1$ by $1$ .

$x = - 1$

Step 6

Evaluate $f (- 1)$ .

Replace the variable $x$ with $- 1$ in the expression.

$f (- 1) = 2 {(- 1)}^{2} + 4 (- 1) - 30$

Simplify the result.

Simplify each term.

Raise $- 1$ to the power of $2$ .

$f (- 1) = 2 \cdot 1 + 4 (- 1) - 30$

Multiply $2$ by $1$ .

$f (- 1) = 2 + 4 (- 1) - 30$

Multiply $4$ by $- 1$ .

$f (- 1) = 2 - 4 - 30$

Simplify by subtracting numbers.

Subtract $4$ from $2$ .

$f (- 1) = - 2 - 30$

Subtract $30$ from $- 2$ .

$f (- 1) = - 32$

The final answer is $- 32$ .

$- 32$

Step 7

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

$(- 1, - 32)$

Step 8

Find the Maximum/Minimum Value y=2(x-3)(x+5)

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