Algebra Examples

$h (r) = (r + 1) (r + 8)$

Step 1

Rewrite the equation in terms of $x$ and $y$ .

$y = (x + 1) (x + 8)$

Step 2

Rewrite the equation in vertex form.

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Step 2.1

Complete the square for $(x + 1) (x + 8)$ .

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Step 2.1.1

Simplify the expression.

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Step 2.1.1.1

Expand $(x + 1) (x + 8)$ using the FOIL Method.

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Step 2.1.1.1.1

Apply the distributive property.

$x (x + 8) + 1 (x + 8)$

Step 2.1.1.1.2

Apply the distributive property.

$x \cdot x + x \cdot 8 + 1 (x + 8)$

Step 2.1.1.1.3

Apply the distributive property.

$x \cdot x + x \cdot 8 + 1 x + 1 \cdot 8$

Step 2.1.1.2

Simplify and combine like terms.

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Step 2.1.1.2.1

Simplify each term.

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Step 2.1.1.2.1.1

Multiply $x$ by $x$ .

$x^{2} + x \cdot 8 + 1 x + 1 \cdot 8$

Step 2.1.1.2.1.2

Move $8$ to the left of $x$ .

$x^{2} + 8 \cdot x + 1 x + 1 \cdot 8$

Step 2.1.1.2.1.3

Multiply $x$ by $1$ .

$x^{2} + 8 x + x + 1 \cdot 8$

Step 2.1.1.2.1.4

Multiply $8$ by $1$ .

$x^{2} + 8 x + x + 8$

Step 2.1.1.2.2

Add $8 x$ and $x$ .

$x^{2} + 9 x + 8$

Step 2.1.2

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

$a = 1$

$b = 9$

$c = 8$

Step 2.1.3

Consider the vertex form of a parabola.

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

Step 2.1.4

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

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Step 2.1.4.1

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

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

Step 2.1.4.2

Multiply $2$ by $1$ .

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

Step 2.1.5

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

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Step 2.1.5.1

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

$e = 8 - \frac{9^{2}}{4 \cdot 1}$

Step 2.1.5.2

Simplify the right side.

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Step 2.1.5.2.1

Simplify each term.

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Step 2.1.5.2.1.1

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

$e = 8 - \frac{81}{4 \cdot 1}$

Step 2.1.5.2.1.2

Multiply $4$ by $1$ .

$e = 8 - \frac{81}{4}$

Step 2.1.5.2.2

To write $8$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$e = 8 \cdot \frac{4}{4} - \frac{81}{4}$

Step 2.1.5.2.3

Combine $8$ and $\frac{4}{4}$ .

$e = \frac{8 \cdot 4}{4} - \frac{81}{4}$

Step 2.1.5.2.4

Combine the numerators over the common denominator.

$e = \frac{8 \cdot 4 - 81}{4}$

Step 2.1.5.2.5

Simplify the numerator.

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Step 2.1.5.2.5.1

Multiply $8$ by $4$ .

$e = \frac{32 - 81}{4}$

Step 2.1.5.2.5.2

Subtract $81$ from $32$ .

$e = \frac{- 49}{4}$

Step 2.1.5.2.6

Move the negative in front of the fraction.

$e = - \frac{49}{4}$

Step 2.1.6

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(x + \frac{9}{2})}^{2} - \frac{49}{4}$ .

${(x + \frac{9}{2})}^{2} - \frac{49}{4}$

Step 2.2

Set $y$ equal to the new right side.

$y = {(x + \frac{9}{2})}^{2} - \frac{49}{4}$

Step 3

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

$a = 1$

$h = - \frac{9}{2}$

$k = - \frac{49}{4}$

Step 4

Find the vertex $(h, k)$ .

$(- \frac{9}{2}, - \frac{49}{4})$

Step 5