Algebra Examples

$f (x) = - 2 x^{2} + 8 x - 7$

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

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Step 2.1

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

$x = - \frac{8}{2 (- 2)}$

Step 2.2

Remove parentheses.

$x = - \frac{8}{2 (- 2)}$

Step 2.3

Simplify $- \frac{8}{2 (- 2)}$ .

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Step 2.3.1

Cancel the common factor of $8$ and $2$ .

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Step 2.3.1.1

Factor $2$ out of $8$ .

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

Step 2.3.1.2

Cancel the common factors.

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Step 2.3.1.2.1

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

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

Step 2.3.1.2.2

Cancel the common factor.

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

Step 2.3.1.2.3

Rewrite the expression.

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

Step 2.3.2

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

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Step 2.3.2.1

Factor $2$ out of $4$ .

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

Step 2.3.2.2

Move the negative one from the denominator of $\frac{2}{- 1}$ .

$x = - (- 1 \cdot 2)$

Step 2.3.3

Multiply $- (- 1 \cdot 2)$ .

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Step 2.3.3.1

Multiply $- 1$ by $2$ .

$x = - - 2$

Step 2.3.3.2

Multiply $- 1$ by $- 2$ .

$x = 2$

Step 3

Evaluate $f (2)$ .

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Step 3.1

Replace the variable $x$ with $2$ in the expression.

$f (2) = - 2 {(2)}^{2} + 8 (2) - 7$

Step 3.2

Simplify the result.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

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

$f (2) = - 2 \cdot 4 + 8 (2) - 7$

Step 3.2.1.2

Multiply $- 2$ by $4$ .

$f (2) = - 8 + 8 (2) - 7$

Step 3.2.1.3

Multiply $8$ by $2$ .

$f (2) = - 8 + 16 - 7$

Step 3.2.2

Simplify by adding and subtracting.

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Step 3.2.2.1

Add $- 8$ and $16$ .

$f (2) = 8 - 7$

Step 3.2.2.2

Subtract $7$ from $8$ .

$f (2) = 1$

Step 3.2.3

The final answer is $1$ .

$1$

Step 4

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

$(2, 1)$

Step 5