Precalculus Examples

$f (x) = 3 x^{2} + 12 x - 3$

Step 1

Write $f (x) = 3 x^{2} + 12 x - 3$ as an equation.

$y = 3 x^{2} + 12 x - 3$

Step 2

Complete the square for $3 x^{2} + 12 x - 3$ .

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Step 2.1

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

$a = 3$

$b = 12$

$c = - 3$

Step 2.2

Consider the vertex form of a parabola.

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

Step 2.3

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

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Step 2.3.1

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

$d = \frac{12}{2 \cdot 3}$

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

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

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Step 2.3.2.1.1

Factor $2$ out of $12$ .

$d = \frac{2 \cdot 6}{2 \cdot 3}$

Step 2.3.2.1.2

Cancel the common factors.

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Step 2.3.2.1.2.1

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

$d = \frac{2 \cdot 6}{2 (3)}$

Step 2.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot 6}{2 \cdot 3}$

Step 2.3.2.1.2.3

Rewrite the expression.

$d = \frac{6}{3}$

Step 2.3.2.2

Cancel the common factor of $6$ and $3$ .

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Step 2.3.2.2.1

Factor $3$ out of $6$ .

$d = \frac{3 \cdot 2}{3}$

Step 2.3.2.2.2

Cancel the common factors.

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Step 2.3.2.2.2.1

Factor $3$ out of $3$ .

$d = \frac{3 \cdot 2}{3 (1)}$

Step 2.3.2.2.2.2

Cancel the common factor.

$d = \frac{3 \cdot 2}{3 \cdot 1}$

Step 2.3.2.2.2.3

Rewrite the expression.

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

Step 2.3.2.2.2.4

Divide $2$ by $1$ .

$d = 2$

Step 2.4

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

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Step 2.4.1

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

$e = - 3 - \frac{12^{2}}{4 \cdot 3}$

Step 2.4.2

Simplify the right side.

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Step 2.4.2.1

Simplify each term.

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Step 2.4.2.1.1

Raise $12$ to the power of $2$ .

$e = - 3 - \frac{144}{4 \cdot 3}$

Step 2.4.2.1.2

Multiply $4$ by $3$ .

$e = - 3 - \frac{144}{12}$

Step 2.4.2.1.3

Divide $144$ by $12$ .

$e = - 3 - 1 \cdot 12$

Step 2.4.2.1.4

Multiply $- 1$ by $12$ .

$e = - 3 - 12$

Step 2.4.2.2

Subtract $12$ from $- 3$ .

$e = - 15$

Step 2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $3 {(x + 2)}^{2} - 15$ .

$3 {(x + 2)}^{2} - 15$

Step 3

Set $y$ equal to the new right side.

$y = 3 {(x + 2)}^{2} - 15$

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