Precalculus Examples

$x^{2} - 3 x - 88 = 0$

Step 1

Add $88$ to both sides of the equation.

$x^{2} - 3 x = 88$

Step 2

To create a trinomial square on the left side of the equation, find a value that is equal to the square of half of $b$ .

${(\frac{b}{2})}^{2} = {(- \frac{3}{2})}^{2}$

Step 3

Add the term to each side of the equation.

$x^{2} - 3 x + {(- \frac{3}{2})}^{2} = 88 + {(- \frac{3}{2})}^{2}$

Step 4

Simplify the equation.

Tap for more steps...

Step 4.1

Simplify the left side.

Tap for more steps...

Step 4.1.1

Simplify each term.

Tap for more steps...

Step 4.1.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Tap for more steps...

Step 4.1.1.1.1

Apply the product rule to $- \frac{3}{2}$ .

$x^{2} - 3 x + {(- 1)}^{2} {(\frac{3}{2})}^{2} = 88 + {(- \frac{3}{2})}^{2}$

Step 4.1.1.1.2

Apply the product rule to $\frac{3}{2}$ .

$x^{2} - 3 x + {(- 1)}^{2} \frac{3^{2}}{2^{2}} = 88 + {(- \frac{3}{2})}^{2}$

Step 4.1.1.2

Raise $- 1$ to the power of $2$ .

$x^{2} - 3 x + 1 \frac{3^{2}}{2^{2}} = 88 + {(- \frac{3}{2})}^{2}$

Step 4.1.1.3

Multiply $\frac{3^{2}}{2^{2}}$ by $1$ .

$x^{2} - 3 x + \frac{3^{2}}{2^{2}} = 88 + {(- \frac{3}{2})}^{2}$

Step 4.1.1.4

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

$x^{2} - 3 x + \frac{9}{2^{2}} = 88 + {(- \frac{3}{2})}^{2}$

Step 4.1.1.5

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

$x^{2} - 3 x + \frac{9}{4} = 88 + {(- \frac{3}{2})}^{2}$

Step 4.2

Simplify the right side.

Tap for more steps...

Step 4.2.1

Simplify $88 + {(- \frac{3}{2})}^{2}$ .

Tap for more steps...

Step 4.2.1.1

Simplify each term.

Tap for more steps...

Step 4.2.1.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Tap for more steps...

Step 4.2.1.1.1.1

Apply the product rule to $- \frac{3}{2}$ .

$x^{2} - 3 x + \frac{9}{4} = 88 + {(- 1)}^{2} {(\frac{3}{2})}^{2}$

Step 4.2.1.1.1.2

Apply the product rule to $\frac{3}{2}$ .

$x^{2} - 3 x + \frac{9}{4} = 88 + {(- 1)}^{2} \frac{3^{2}}{2^{2}}$

Step 4.2.1.1.2

Raise $- 1$ to the power of $2$ .

$x^{2} - 3 x + \frac{9}{4} = 88 + 1 \frac{3^{2}}{2^{2}}$

Step 4.2.1.1.3

Multiply $\frac{3^{2}}{2^{2}}$ by $1$ .

$x^{2} - 3 x + \frac{9}{4} = 88 + \frac{3^{2}}{2^{2}}$

Step 4.2.1.1.4

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

$x^{2} - 3 x + \frac{9}{4} = 88 + \frac{9}{2^{2}}$

Step 4.2.1.1.5

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

$x^{2} - 3 x + \frac{9}{4} = 88 + \frac{9}{4}$

Step 4.2.1.2

To write $88$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x^{2} - 3 x + \frac{9}{4} = 88 \cdot \frac{4}{4} + \frac{9}{4}$

Step 4.2.1.3

Combine $88$ and $\frac{4}{4}$ .

$x^{2} - 3 x + \frac{9}{4} = \frac{88 \cdot 4}{4} + \frac{9}{4}$

Step 4.2.1.4

Combine the numerators over the common denominator.

$x^{2} - 3 x + \frac{9}{4} = \frac{88 \cdot 4 + 9}{4}$

Step 4.2.1.5

Simplify the numerator.

Tap for more steps...

Step 4.2.1.5.1

Multiply $88$ by $4$ .

$x^{2} - 3 x + \frac{9}{4} = \frac{352 + 9}{4}$

Step 4.2.1.5.2

Add $352$ and $9$ .

$x^{2} - 3 x + \frac{9}{4} = \frac{361}{4}$

Step 5

Factor the perfect trinomial square into ${(x - \frac{3}{2})}^{2}$ .

${(x - \frac{3}{2})}^{2} = \frac{361}{4}$

Step 6

Solve the equation for $x$ .

Tap for more steps...

Step 6.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x - \frac{3}{2} = \pm \sqrt{\frac{361}{4}}$

Step 6.2

Simplify $\pm \sqrt{\frac{361}{4}}$ .

Tap for more steps...

Step 6.2.1

Rewrite $\sqrt{\frac{361}{4}}$ as $\frac{\sqrt{361}}{\sqrt{4}}$ .

$x - \frac{3}{2} = \pm \frac{\sqrt{361}}{\sqrt{4}}$

Step 6.2.2

Simplify the numerator.

Tap for more steps...

Step 6.2.2.1

Rewrite $361$ as $19^{2}$ .

$x - \frac{3}{2} = \pm \frac{\sqrt{19^{2}}}{\sqrt{4}}$

Step 6.2.2.2

Pull terms out from under the radical, assuming positive real numbers.

$x - \frac{3}{2} = \pm \frac{19}{\sqrt{4}}$

Step 6.2.3

Simplify the denominator.

Tap for more steps...

Step 6.2.3.1

Rewrite $4$ as $2^{2}$ .

$x - \frac{3}{2} = \pm \frac{19}{\sqrt{2^{2}}}$

Step 6.2.3.2

Pull terms out from under the radical, assuming positive real numbers.

$x - \frac{3}{2} = \pm \frac{19}{2}$

Step 6.3

The complete solution is the result of both the positive and negative portions of the solution.

Tap for more steps...

Step 6.3.1

First, use the positive value of the $\pm$ to find the first solution.

$x - \frac{3}{2} = \frac{19}{2}$

Step 6.3.2

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

Tap for more steps...

Step 6.3.2.1

Add $\frac{3}{2}$ to both sides of the equation.

$x = \frac{19}{2} + \frac{3}{2}$

Step 6.3.2.2

Combine the numerators over the common denominator.

$x = \frac{19 + 3}{2}$

Step 6.3.2.3

Add $19$ and $3$ .

$x = \frac{22}{2}$

Step 6.3.2.4

Divide $22$ by $2$ .

$x = 11$

Step 6.3.3

Next, use the negative value of the $\pm$ to find the second solution.

$x - \frac{3}{2} = - \frac{19}{2}$

Step 6.3.4

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

Tap for more steps...

Step 6.3.4.1

Add $\frac{3}{2}$ to both sides of the equation.

$x = - \frac{19}{2} + \frac{3}{2}$

Step 6.3.4.2

Combine the numerators over the common denominator.

$x = \frac{- 19 + 3}{2}$

Step 6.3.4.3

Add $- 19$ and $3$ .

$x = \frac{- 16}{2}$

Step 6.3.4.4

Divide $- 16$ by $2$ .

$x = - 8$

Step 6.3.5

The complete solution is the result of both the positive and negative portions of the solution.

$x = 11, - 8$