Find the Vertex Form y=(x+5)(x+4)

$y = (x + 5) (x + 4)$

Step 1

Complete the square for $(x + 5) (x + 4)$ .

Simplify the expression.

Expand $(x + 5) (x + 4)$ using the FOIL Method.

Apply the distributive property.

$x (x + 4) + 5 (x + 4)$

Apply the distributive property.

$x \cdot x + x \cdot 4 + 5 (x + 4)$

Apply the distributive property.

$x \cdot x + x \cdot 4 + 5 x + 5 \cdot 4$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$x^{2} + x \cdot 4 + 5 x + 5 \cdot 4$

Move $4$ to the left of $x$ .

$x^{2} + 4 \cdot x + 5 x + 5 \cdot 4$

Multiply $5$ by $4$ .

$x^{2} + 4 x + 5 x + 20$

Add $4 x$ and $5 x$ .

$x^{2} + 9 x + 20$

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

$a = 1$

$b = 9$

$c = 20$

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 $d = \frac{b}{2 a}$ .

$d = \frac{9}{2 \cdot 1}$

Multiply $2$ by $1$ .

$d = \frac{9}{2}$

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 $e = c - \frac{b^{2}}{4 a}$ .

$e = 20 - \frac{9^{2}}{4 \cdot 1}$

Simplify the right side.

Simplify each term.

Raise $9$ to the power of $2$ .

$e = 20 - \frac{81}{4 \cdot 1}$

Multiply $4$ by $1$ .

$e = 20 - \frac{81}{4}$

To write $20$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$e = 20 \cdot \frac{4}{4} - \frac{81}{4}$

Combine $20$ and $\frac{4}{4}$ .

$e = \frac{20 \cdot 4}{4} - \frac{81}{4}$

Combine the numerators over the common denominator.

$e = \frac{20 \cdot 4 - 81}{4}$

Simplify the numerator.

Multiply $20$ by $4$ .

$e = \frac{80 - 81}{4}$

Subtract $81$ from $80$ .

$e = \frac{- 1}{4}$

Move the negative in front of the fraction.

$e = - \frac{1}{4}$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(x + \frac{9}{2})}^{2} - \frac{1}{4}$ .

${(x + \frac{9}{2})}^{2} - \frac{1}{4}$

Step 2

Set $y$ equal to the new right side.

$y = {(x + \frac{9}{2})}^{2} - \frac{1}{4}$

Find the Vertex Form y=(x+5)(x+4)

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