Algebra Examples

$y = x^{2} + x - 12$

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.

$x^{2} + x - 12 = y$

Step 2

Add $12$ to both sides of the equation.

$x^{2} + x = y + 12$

Step 3

Complete the square.

Tap for more steps...

Step 3.1

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

$a = 1$

$b = 1$

$c = 0 = y + 12$

Step 3.2

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e = y + 12$

Step 3.3

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

Tap for more steps...

Step 3.3.1

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

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

Step 3.3.2

Cancel the common factor of $1$ .

Tap for more steps...

Step 3.3.2.1

Cancel the common factor.

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

Step 3.3.2.2

Rewrite the expression.

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

Step 3.4

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

Tap for more steps...

Step 3.4.1

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

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

Step 3.4.2

Simplify the right side.

Tap for more steps...

Step 3.4.2.1

Simplify each term.

Tap for more steps...

Step 3.4.2.1.1

One to any power is one.

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

Step 3.4.2.1.2

Multiply $4$ by $1$ .

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

Step 3.4.2.2

Subtract $\frac{1}{4}$ from $0$ .

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

Step 3.5

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

${(x + \frac{1}{2})}^{2} - \frac{1}{4} = y + 12$

Step 4

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

Tap for more steps...

Step 4.1

Add $\frac{1}{4}$ to both sides of the equation.

${(x + \frac{1}{2})}^{2} = y + 12 + \frac{1}{4}$

Step 4.2

To write $12$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

${(x + \frac{1}{2})}^{2} = y + 12 \cdot \frac{4}{4} + \frac{1}{4}$

Step 4.3

Combine $12$ and $\frac{4}{4}$ .

${(x + \frac{1}{2})}^{2} = y + \frac{12 \cdot 4}{4} + \frac{1}{4}$

Step 4.4

Combine the numerators over the common denominator.

${(x + \frac{1}{2})}^{2} = y + \frac{12 \cdot 4 + 1}{4}$

Step 4.5

Simplify the numerator.

Tap for more steps...

Step 4.5.1

Multiply $12$ by $4$ .

${(x + \frac{1}{2})}^{2} = y + \frac{48 + 1}{4}$

Step 4.5.2

Add $48$ and $1$ .

${(x + \frac{1}{2})}^{2} = y + \frac{49}{4}$

Step 5

Factor.

Tap for more steps...

Step 5.1

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

${(x + \frac{1}{2})}^{2} = y \cdot \frac{4}{4} + \frac{49}{4}$

Step 5.2

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

${(x + \frac{1}{2})}^{2} = \frac{y \cdot 4}{4} + \frac{49}{4}$

Step 5.3

Combine the numerators over the common denominator.

${(x + \frac{1}{2})}^{2} = \frac{y \cdot 4 + 49}{4}$

Step 5.4

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

${(x + \frac{1}{2})}^{2} = \frac{4 y + 49}{4}$

Step 5.5

Reorder terms.

${(x + \frac{1}{2})}^{2} = \frac{1}{4} (4 y + 49)$