Algebra Examples

$y = x^{2} - 9 x + 18$

Step 1

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$x^{2} - 9 x + 18 = y$

Step 2

Subtract $18$ from both sides of the equation.

$x^{2} - 9 x = y - 18$

Step 3

Complete the square.

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Step 3.1

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

$a = 1$

$b = - 9$

$c = 0 = y - 18$

Step 3.2

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e = y - 18$

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Multiply $2$ by $1$ .

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

Step 3.3.2.2

Move the negative in front of the fraction.

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

Step 3.4

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

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Step 3.4.1

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

$e = 0 - \frac{{(- 9)}^{2}}{4 \cdot 1}$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify each term.

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Step 3.4.2.1.1

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

$e = 0 - \frac{81}{4 \cdot 1}$

Step 3.4.2.1.2

Multiply $4$ by $1$ .

$e = 0 - \frac{81}{4}$

Step 3.4.2.2

Subtract $\frac{81}{4}$ from $0$ .

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

Step 3.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(x - \frac{9}{2})}^{2} - \frac{81}{4}$ .

${(x - \frac{9}{2})}^{2} - \frac{81}{4} = y - 18$

Step 4

Move all terms not containing $x$ to the right side of the equation.

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Step 4.1

Add $\frac{81}{4}$ to both sides of the equation.

${(x - \frac{9}{2})}^{2} = y - 18 + \frac{81}{4}$

Step 4.2

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

${(x - \frac{9}{2})}^{2} = y - 18 \cdot \frac{4}{4} + \frac{81}{4}$

Step 4.3

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

${(x - \frac{9}{2})}^{2} = y + \frac{- 18 \cdot 4}{4} + \frac{81}{4}$

Step 4.4

Combine the numerators over the common denominator.

${(x - \frac{9}{2})}^{2} = y + \frac{- 18 \cdot 4 + 81}{4}$

Step 4.5

Simplify the numerator.

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Step 4.5.1

Multiply $- 18$ by $4$ .

${(x - \frac{9}{2})}^{2} = y + \frac{- 72 + 81}{4}$

Step 4.5.2

Add $- 72$ and $81$ .

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

Step 5

Factor.

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

Combine the numerators over the common denominator.

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

Step 5.4

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

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

Step 5.5

Reorder terms.

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