Find the Maximum/Minimum Value y=-3x^2-42x-150

$y = - 3 x^{2} - 42 x - 150$

Step 1

The maximum of a quadratic function occurs at $x = - \frac{b}{2 a}$ . If $a$ is negative, the maximum value of the function is $f (- \frac{b}{2 a})$ .

$f_{max}$ $x = a x^{2} + b x + c$ 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 $b$ .

$x = - \frac{- 42}{2 (- 3)}$

Remove parentheses.

$x = - \frac{- 42}{2 (- 3)}$

Simplify $- \frac{- 42}{2 (- 3)}$ .

Cancel the common factor of $- 42$ and $2$ .

Factor $2$ out of $- 42$ .

$x = - \frac{2 \cdot - 21}{2 \cdot - 3}$

Cancel the common factors.

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

$x = - \frac{2 \cdot - 21}{2 (- 3)}$

Cancel the common factor.

$x = - \frac{2 \cdot - 21}{2 \cdot - 3}$

Rewrite the expression.

$x = - \frac{- 21}{- 3}$

Cancel the common factor of $- 21$ and $- 3$ .

Factor $- 3$ out of $- 21$ .

$x = - \frac{- 3 (7)}{- 3}$

Cancel the common factors.

Factor $- 3$ out of $- 3$ .

$x = - \frac{- 3 \cdot 7}{- 3 \cdot 1}$

Cancel the common factor.

$x = - \frac{- 3 \cdot 7}{- 3 \cdot 1}$

Rewrite the expression.

$x = - \frac{7}{1}$

Divide $7$ by $1$ .

$x = - 1 \cdot 7$

Multiply $- 1$ by $7$ .

$x = - 7$

Step 3

Evaluate $f (- 7)$ .

Replace the variable $x$ with $- 7$ in the expression.

$f (- 7) = - 3 {(- 7)}^{2} - 42 \cdot - 7 - 150$

Simplify the result.

Simplify each term.

Raise $- 7$ to the power of $2$ .

$f (- 7) = - 3 \cdot 49 - 42 \cdot - 7 - 150$

Multiply $- 3$ by $49$ .

$f (- 7) = - 147 - 42 \cdot - 7 - 150$

Multiply $- 42$ by $- 7$ .

$f (- 7) = - 147 + 294 - 150$

Simplify by adding and subtracting.

Add $- 147$ and $294$ .

$f (- 7) = 147 - 150$

Subtract $150$ from $147$ .

$f (- 7) = - 3$

The final answer is $- 3$ .

$- 3$

Step 4

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

$(- 7, - 3)$

Step 5

Find the Maximum/Minimum Value y=-3x^2-42x-150

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