Find the Vertex Form (x-3)(x+6)

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

Step 1

Write $(x - 3) (x + 6)$ as an equation.

$y = (x - 3) (x + 6)$

Step 2

Complete the square for $(x - 3) (x + 6)$ .

Simplify the expression.

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

Move $6$ to the left of $x$ .

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

Multiply $- 3$ by $6$ .

$x^{2} + 6 x - 3 x - 18$

Subtract $3 x$ from $6 x$ .

$x^{2} + 3 x - 18$

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

$a = 1$

$b = 3$

$c = - 18$

Consider the vertex form of a parabola.

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

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

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

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

Multiply $2$ by $1$ .

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

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

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

$e = - 18 - \frac{3^{2}}{4 \cdot 1}$

Simplify the right side.

Simplify each term.

Raise $3$ to the power of $2$ .

$e = - 18 - \frac{9}{4 \cdot 1}$

Multiply $4$ by $1$ .

$e = - 18 - \frac{9}{4}$

To write $- 18$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$e = - 18 \cdot \frac{4}{4} - \frac{9}{4}$

Combine $- 18$ and $\frac{4}{4}$ .

$e = \frac{- 18 \cdot 4}{4} - \frac{9}{4}$

Combine the numerators over the common denominator.

$e = \frac{- 18 \cdot 4 - 9}{4}$

Simplify the numerator.

Multiply $- 18$ by $4$ .

$e = \frac{- 72 - 9}{4}$

Subtract $9$ from $- 72$ .

$e = \frac{- 81}{4}$

Move the negative in front of the fraction.

$e = - \frac{81}{4}$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(x + \frac{3}{2})}^{2} - \frac{81}{4}$ .

${(x + \frac{3}{2})}^{2} - \frac{81}{4}$

Step 3

Set $y$ equal to the new right side.

$y = {(x + \frac{3}{2})}^{2} - \frac{81}{4}$

Find the Vertex Form (x-3)(x+6)

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