Algebra Examples

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

Step 1

Write $f (x) = 8 x - 4 + 2 x^{2}$ as an equation.

$y = 8 x - 4 + 2 x^{2}$

Step 2

Simplify $8 x - 4 + 2 x^{2}$ .

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Step 2.1

Move $- 4$ .

$y = 8 x + 2 x^{2} - 4$

Step 2.2

Reorder $8 x$ and $2 x^{2}$ .

$y = 2 x^{2} + 8 x - 4$

Step 3

Complete the square for $2 x^{2} + 8 x - 4$ .

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Step 3.1

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = 2$

$b = 8$

$c = - 4$

Step 3.2

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Step 3.3

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

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Step 3.3.1

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{8}{2 \cdot 2}$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

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

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Step 3.3.2.1.1

Factor $2$ out of $8$ .

$d = \frac{2 \cdot 4}{2 \cdot 2}$

Step 3.3.2.1.2

Cancel the common factors.

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Step 3.3.2.1.2.1

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

$d = \frac{2 \cdot 4}{2 (2)}$

Step 3.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot 4}{2 \cdot 2}$

Step 3.3.2.1.2.3

Rewrite the expression.

$d = \frac{4}{2}$

Step 3.3.2.2

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

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Step 3.3.2.2.1

Factor $2$ out of $4$ .

$d = \frac{2 \cdot 2}{2}$

Step 3.3.2.2.2

Cancel the common factors.

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Step 3.3.2.2.2.1

Factor $2$ out of $2$ .

$d = \frac{2 \cdot 2}{2 (1)}$

Step 3.3.2.2.2.2

Cancel the common factor.

$d = \frac{2 \cdot 2}{2 \cdot 1}$

Step 3.3.2.2.2.3

Rewrite the expression.

$d = \frac{2}{1}$

Step 3.3.2.2.2.4

Divide $2$ by $1$ .

$d = 2$

Step 3.4

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

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Step 3.4.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = - 4 - \frac{8^{2}}{4 \cdot 2}$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify each term.

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Step 3.4.2.1.1

Raise $8$ to the power of $2$ .

$e = - 4 - \frac{64}{4 \cdot 2}$

Step 3.4.2.1.2

Multiply $4$ by $2$ .

$e = - 4 - \frac{64}{8}$

Step 3.4.2.1.3

Divide $64$ by $8$ .

$e = - 4 - 1 \cdot 8$

Step 3.4.2.1.4

Multiply $- 1$ by $8$ .

$e = - 4 - 8$

Step 3.4.2.2

Subtract $8$ from $- 4$ .

$e = - 12$

Step 3.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $2 {(x + 2)}^{2} - 12$ .

$2 {(x + 2)}^{2} - 12$

Step 4

Set $y$ equal to the new right side.

$y = 2 {(x + 2)}^{2} - 12$

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