Algebra Examples

$f (x) = 3 (x + 2) (x - 2)$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

Complete the square for $3 (x + 2) (x - 2)$ .

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Step 1.1.1

Simplify the expression.

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Step 1.1.1.1

Apply the distributive property.

$(3 x + 3 \cdot 2) (x - 2)$

Step 1.1.1.2

Multiply $3$ by $2$ .

$(3 x + 6) (x - 2)$

Step 1.1.1.3

Expand $(3 x + 6) (x - 2)$ using the FOIL Method.

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Step 1.1.1.3.1

Apply the distributive property.

$3 x (x - 2) + 6 (x - 2)$

Step 1.1.1.3.2

Apply the distributive property.

$3 x \cdot x + 3 x \cdot - 2 + 6 (x - 2)$

Step 1.1.1.3.3

Apply the distributive property.

$3 x \cdot x + 3 x \cdot - 2 + 6 x + 6 \cdot - 2$

Step 1.1.1.4

Simplify and combine like terms.

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Step 1.1.1.4.1

Simplify each term.

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Step 1.1.1.4.1.1

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

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Step 1.1.1.4.1.1.1

Move $x$ .

$3 (x \cdot x) + 3 x \cdot - 2 + 6 x + 6 \cdot - 2$

Step 1.1.1.4.1.1.2

Multiply $x$ by $x$ .

$3 x^{2} + 3 x \cdot - 2 + 6 x + 6 \cdot - 2$

Step 1.1.1.4.1.2

Multiply $- 2$ by $3$ .

$3 x^{2} - 6 x + 6 x + 6 \cdot - 2$

Step 1.1.1.4.1.3

Multiply $6$ by $- 2$ .

$3 x^{2} - 6 x + 6 x - 12$

Step 1.1.1.4.2

Add $- 6 x$ and $6 x$ .

$3 x^{2} + 0 - 12$

Step 1.1.1.4.3

Add $3 x^{2}$ and $0$ .

$3 x^{2} - 12$

Step 1.1.2

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

$a = 3$

$b = 0$

$c = - 12$

Step 1.1.3

Consider the vertex form of a parabola.

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

Step 1.1.4

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

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Step 1.1.4.1

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

$d = \frac{0}{2 \cdot 3}$

Step 1.1.4.2

Simplify the right side.

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Step 1.1.4.2.1

Cancel the common factor of $0$ and $2$ .

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Step 1.1.4.2.1.1

Factor $2$ out of $0$ .

$d = \frac{2 (0)}{2 \cdot 3}$

Step 1.1.4.2.1.2

Cancel the common factors.

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Step 1.1.4.2.1.2.1

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

$d = \frac{2 (0)}{2 (3)}$

Step 1.1.4.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot 0}{2 \cdot 3}$

Step 1.1.4.2.1.2.3

Rewrite the expression.

$d = \frac{0}{3}$

Step 1.1.4.2.2

Cancel the common factor of $0$ and $3$ .

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Step 1.1.4.2.2.1

Factor $3$ out of $0$ .

$d = \frac{3 (0)}{3}$

Step 1.1.4.2.2.2

Cancel the common factors.

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Step 1.1.4.2.2.2.1

Factor $3$ out of $3$ .

$d = \frac{3 \cdot 0}{3 \cdot 1}$

Step 1.1.4.2.2.2.2

Cancel the common factor.

$d = \frac{3 \cdot 0}{3 \cdot 1}$

Step 1.1.4.2.2.2.3

Rewrite the expression.

$d = \frac{0}{1}$

Step 1.1.4.2.2.2.4

Divide $0$ by $1$ .

$d = 0$

Step 1.1.5

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

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Step 1.1.5.1

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

$e = - 12 - \frac{0^{2}}{4 \cdot 3}$

Step 1.1.5.2

Simplify the right side.

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Step 1.1.5.2.1

Simplify each term.

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Step 1.1.5.2.1.1

Raising $0$ to any positive power yields $0$ .

$e = - 12 - \frac{0}{4 \cdot 3}$

Step 1.1.5.2.1.2

Multiply $4$ by $3$ .

$e = - 12 - \frac{0}{12}$

Step 1.1.5.2.1.3

Divide $0$ by $12$ .

$e = - 12 - 0$

Step 1.1.5.2.1.4

Multiply $- 1$ by $0$ .

$e = - 12 + 0$

Step 1.1.5.2.2

Add $- 12$ and $0$ .

$e = - 12$

Step 1.1.6

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $3 {(x + 0)}^{2} - 12$ .

$3 {(x + 0)}^{2} - 12$

Step 1.2

Set $y$ equal to the new right side.

$y = 3 {(x + 0)}^{2} - 12$

Step 2

Use the vertex form, $y = a {(x - h)}^{2} + k$ , to determine the values of $a$ , $h$ , and $k$ .

$a = 3$

$h = 0$

$k = - 12$

Step 3

Find the vertex $(h, k)$ .

$(0, - 12)$

Step 4