Precalculus Examples

$x^{2} - \frac{3}{4} x = \frac{1}{8}$

Step 1

Move all terms to the left side of the equation and simplify.

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Step 1.1

Simplify the left side.

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Step 1.1.1

Simplify each term.

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Step 1.1.1.1

Combine $x$ and $\frac{3}{4}$ .

$x^{2} - \frac{x \cdot 3}{4} = \frac{1}{8}$

Step 1.1.1.2

Move $3$ to the left of $x$ .

$x^{2} - \frac{3 x}{4} = \frac{1}{8}$

Step 1.2

Subtract $\frac{1}{8}$ from both sides of the equation.

$x^{2} - \frac{3 x}{4} - \frac{1}{8} = 0$

Step 2

Multiply through by the least common denominator $8$ , then simplify.

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Step 2.1

Apply the distributive property.

$8 x^{2} + 8 (- \frac{3 x}{4}) + 8 (- \frac{1}{8}) = 0$

Step 2.2

Simplify.

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Step 2.2.1

Cancel the common factor of $4$ .

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Step 2.2.1.1

Move the leading negative in $- \frac{3 x}{4}$ into the numerator.

$8 x^{2} + 8 (\frac{- 3 x}{4}) + 8 (- \frac{1}{8}) = 0$

Step 2.2.1.2

Factor $4$ out of $8$ .

$8 x^{2} + 4 (2) (\frac{- 3 x}{4}) + 8 (- \frac{1}{8}) = 0$

Step 2.2.1.3

Cancel the common factor.

$8 x^{2} + 4 \cdot (2 (\frac{- 3 x}{4})) + 8 (- \frac{1}{8}) = 0$

Step 2.2.1.4

Rewrite the expression.

$8 x^{2} + 2 (- 3 x) + 8 (- \frac{1}{8}) = 0$

Step 2.2.2

Multiply $- 3$ by $2$ .

$8 x^{2} - 6 x + 8 (- \frac{1}{8}) = 0$

Step 2.2.3

Cancel the common factor of $8$ .

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Step 2.2.3.1

Move the leading negative in $- \frac{1}{8}$ into the numerator.

$8 x^{2} - 6 x + 8 (\frac{- 1}{8}) = 0$

Step 2.2.3.2

Cancel the common factor.

$8 x^{2} - 6 x + 8 (\frac{- 1}{8}) = 0$

Step 2.2.3.3

Rewrite the expression.

$8 x^{2} - 6 x - 1 = 0$

Step 3

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 4

Substitute the values $a = 8$ , $b = - 6$ , and $c = - 1$ into the quadratic formula and solve for $x$ .

$\frac{6 \pm \sqrt{{(- 6)}^{2} - 4 \cdot (8 \cdot - 1)}}{2 \cdot 8}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $- 6$ to the power of $2$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 8 \cdot - 1}}{2 \cdot 8}$

Step 5.1.2

Multiply $- 4 \cdot 8 \cdot - 1$ .

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Step 5.1.2.1

Multiply $- 4$ by $8$ .

$x = \frac{6 \pm \sqrt{36 - 32 \cdot - 1}}{2 \cdot 8}$

Step 5.1.2.2

Multiply $- 32$ by $- 1$ .

$x = \frac{6 \pm \sqrt{36 + 32}}{2 \cdot 8}$

Step 5.1.3

Add $36$ and $32$ .

$x = \frac{6 \pm \sqrt{68}}{2 \cdot 8}$

Step 5.1.4

Rewrite $68$ as $2^{2} \cdot 17$ .

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Step 5.1.4.1

Factor $4$ out of $68$ .

$x = \frac{6 \pm \sqrt{4 (17)}}{2 \cdot 8}$

Step 5.1.4.2

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

$x = \frac{6 \pm \sqrt{2^{2} \cdot 17}}{2 \cdot 8}$

Step 5.1.5

Pull terms out from under the radical.

$x = \frac{6 \pm 2 \sqrt{17}}{2 \cdot 8}$

Step 5.2

Multiply $2$ by $8$ .

$x = \frac{6 \pm 2 \sqrt{17}}{16}$

Step 5.3

Simplify $\frac{6 \pm 2 \sqrt{17}}{16}$ .

$x = \frac{3 \pm \sqrt{17}}{8}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{3 \pm \sqrt{17}}{8}$

Decimal Form:

$x = 0.89038820 \dots, - 0.14038820 \dots$