Finite Math Examples

$x^{2} - 3 x + 15 = 3$

Step 1

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

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Step 1.1

Subtract $15$ from both sides of the equation.

$x^{2} - 3 x = 3 - 15$

Step 1.2

Subtract $15$ from $3$ .

$x^{2} - 3 x = - 12$

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.

$x^{2} - 3 x + {(- \frac{3}{2})}^{2} = - 12 + {(- \frac{3}{2})}^{2}$

Step 4

Simplify the equation.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Simplify each term.

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Step 4.1.1.1

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

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Step 4.1.1.1.1

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

$x^{2} - 3 x + {(- 1)}^{2} {(\frac{3}{2})}^{2} = - 12 + {(- \frac{3}{2})}^{2}$

Step 4.1.1.1.2

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

$x^{2} - 3 x + {(- 1)}^{2} \frac{3^{2}}{2^{2}} = - 12 + {(- \frac{3}{2})}^{2}$

Step 4.1.1.2

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

$x^{2} - 3 x + 1 \frac{3^{2}}{2^{2}} = - 12 + {(- \frac{3}{2})}^{2}$

Step 4.1.1.3

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

$x^{2} - 3 x + \frac{3^{2}}{2^{2}} = - 12 + {(- \frac{3}{2})}^{2}$

Step 4.1.1.4

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

$x^{2} - 3 x + \frac{9}{2^{2}} = - 12 + {(- \frac{3}{2})}^{2}$

Step 4.1.1.5

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

$x^{2} - 3 x + \frac{9}{4} = - 12 + {(- \frac{3}{2})}^{2}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $- 12 + {(- \frac{3}{2})}^{2}$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

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

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Step 4.2.1.1.1.1

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

$x^{2} - 3 x + \frac{9}{4} = - 12 + {(- 1)}^{2} {(\frac{3}{2})}^{2}$

Step 4.2.1.1.1.2

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

$x^{2} - 3 x + \frac{9}{4} = - 12 + {(- 1)}^{2} \frac{3^{2}}{2^{2}}$

Step 4.2.1.1.2

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

$x^{2} - 3 x + \frac{9}{4} = - 12 + 1 \frac{3^{2}}{2^{2}}$

Step 4.2.1.1.3

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

$x^{2} - 3 x + \frac{9}{4} = - 12 + \frac{3^{2}}{2^{2}}$

Step 4.2.1.1.4

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

$x^{2} - 3 x + \frac{9}{4} = - 12 + \frac{9}{2^{2}}$

Step 4.2.1.1.5

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

$x^{2} - 3 x + \frac{9}{4} = - 12 + \frac{9}{4}$

Step 4.2.1.2

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

$x^{2} - 3 x + \frac{9}{4} = - 12 \cdot \frac{4}{4} + \frac{9}{4}$

Step 4.2.1.3

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

$x^{2} - 3 x + \frac{9}{4} = \frac{- 12 \cdot 4}{4} + \frac{9}{4}$

Step 4.2.1.4

Combine the numerators over the common denominator.

$x^{2} - 3 x + \frac{9}{4} = \frac{- 12 \cdot 4 + 9}{4}$

Step 4.2.1.5

Simplify the numerator.

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Step 4.2.1.5.1

Multiply $- 12$ by $4$ .

$x^{2} - 3 x + \frac{9}{4} = \frac{- 48 + 9}{4}$

Step 4.2.1.5.2

Add $- 48$ and $9$ .

$x^{2} - 3 x + \frac{9}{4} = \frac{- 39}{4}$

Step 4.2.1.6

Move the negative in front of the fraction.

$x^{2} - 3 x + \frac{9}{4} = - \frac{39}{4}$

Step 5

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

${(x - \frac{3}{2})}^{2} = - \frac{39}{4}$

Step 6

Solve the equation for $x$ .

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Step 6.1

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

$x - \frac{3}{2} = \pm \sqrt{- \frac{39}{4}}$

Step 6.2

Simplify $\pm \sqrt{- \frac{39}{4}}$ .

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Step 6.2.1

Rewrite $- \frac{39}{4}$ as ${(\frac{i}{2})}^{2} \cdot 39$ .

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Step 6.2.1.1

Rewrite $- 1$ as $i^{2}$ .

$x - \frac{3}{2} = \pm \sqrt{i^{2} \frac{39}{4}}$

Step 6.2.1.2

Factor the perfect power $1^{2}$ out of $39$ .

$x - \frac{3}{2} = \pm \sqrt{i^{2} \frac{1^{2} \cdot 39}{4}}$

Step 6.2.1.3

Factor the perfect power $2^{2}$ out of $4$ .

$x - \frac{3}{2} = \pm \sqrt{i^{2} \frac{1^{2} \cdot 39}{2^{2} \cdot 1}}$

Step 6.2.1.4

Rearrange the fraction $\frac{1^{2} \cdot 39}{2^{2} \cdot 1}$ .

$x - \frac{3}{2} = \pm \sqrt{i^{2} ({(\frac{1}{2})}^{2} \cdot 39)}$

Step 6.2.1.5

Rewrite $i^{2} {(\frac{1}{2})}^{2}$ as ${(\frac{i}{2})}^{2}$ .

$x - \frac{3}{2} = \pm \sqrt{{(\frac{i}{2})}^{2} \cdot 39}$

Step 6.2.2

Pull terms out from under the radical.

$x - \frac{3}{2} = \pm \frac{i}{2} \sqrt{39}$

Step 6.2.3

Combine $\frac{i}{2}$ and $\sqrt{39}$ .

$x - \frac{3}{2} = \pm \frac{i \sqrt{39}}{2}$

Step 6.3

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

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Step 6.3.1

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

$x = \pm \frac{i \sqrt{39}}{2} + \frac{3}{2}$

Step 6.3.2

Reorder $\pm \frac{i \sqrt{39}}{2}$ and $\frac{3}{2}$ .

$x = \frac{3}{2} \pm \frac{i \sqrt{39}}{2}$