Algebra Examples

$x^{2} - \frac{1}{4} x - \frac{15}{64} = 0$

Step 1

Combine $x$ and $\frac{1}{4}$ .

$x^{2} - \frac{x}{4} - \frac{15}{64} = 0$

Step 2

Multiply through by the least common denominator $64$ , then simplify.

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Step 2.1

Apply the distributive property.

$64 x^{2} + 64 (- \frac{x}{4}) + 64 (- \frac{15}{64}) = 0$

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 $- \frac{x}{4}$ into the numerator.

$64 x^{2} + 64 (\frac{- x}{4}) + 64 (- \frac{15}{64}) = 0$

Step 2.2.1.2

Factor $4$ out of $64$ .

$64 x^{2} + 4 (16) (\frac{- x}{4}) + 64 (- \frac{15}{64}) = 0$

Step 2.2.1.3

Cancel the common factor.

$64 x^{2} + 4 \cdot (16 (\frac{- x}{4})) + 64 (- \frac{15}{64}) = 0$

Step 2.2.1.4

Rewrite the expression.

$64 x^{2} + 16 (- x) + 64 (- \frac{15}{64}) = 0$

Step 2.2.2

Multiply $- 1$ by $16$ .

$64 x^{2} - 16 x + 64 (- \frac{15}{64}) = 0$

Step 2.2.3

Cancel the common factor of $64$ .

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Step 2.2.3.1

Move the leading negative in $- \frac{15}{64}$ into the numerator.

$64 x^{2} - 16 x + 64 (\frac{- 15}{64}) = 0$

Step 2.2.3.2

Cancel the common factor.

$64 x^{2} - 16 x + 64 (\frac{- 15}{64}) = 0$

Step 2.2.3.3

Rewrite the expression.

$64 x^{2} - 16 x - 15 = 0$

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 $a = 64$ , $b = - 16$ , and $c = - 15$ into the quadratic formula and solve for $x$ .

$\frac{16 \pm \sqrt{{(- 16)}^{2} - 4 \cdot (64 \cdot - 15)}}{2 \cdot 64}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $- 16$ to the power of $2$ .

$x = \frac{16 \pm \sqrt{256 - 4 \cdot 64 \cdot - 15}}{2 \cdot 64}$

Step 5.1.2

Multiply $- 4 \cdot 64 \cdot - 15$ .

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Step 5.1.2.1

Multiply $- 4$ by $64$ .

$x = \frac{16 \pm \sqrt{256 - 256 \cdot - 15}}{2 \cdot 64}$

Step 5.1.2.2

Multiply $- 256$ by $- 15$ .

$x = \frac{16 \pm \sqrt{256 + 3840}}{2 \cdot 64}$

Step 5.1.3

Add $256$ and $3840$ .

$x = \frac{16 \pm \sqrt{4096}}{2 \cdot 64}$

Step 5.1.4

Rewrite $4096$ as $64^{2}$ .

$x = \frac{16 \pm \sqrt{64^{2}}}{2 \cdot 64}$

Step 5.1.5

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

$x = \frac{16 \pm 64}{2 \cdot 64}$

Step 5.2

Multiply $2$ by $64$ .

$x = \frac{16 \pm 64}{128}$

Step 5.3

Simplify $\frac{16 \pm 64}{128}$ .

$x = \frac{1 \pm 4}{8}$

Step 6

The final answer is the combination of both solutions.

$x = \frac{5}{8}, - \frac{3}{8}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \frac{5}{8}, - \frac{3}{8}$

Decimal Form:

$x = 0.625, - 0.375$