Finite Math Examples

$\frac{1}{16} x^{2} - \frac{1}{8} x = 1$

Step 1

Simplify each term.

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Step 1.1

Combine $\frac{1}{16}$ and $x^{2}$ .

$\frac{x^{2}}{16} - \frac{1}{8} x = 1$

Step 1.2

Combine $x$ and $\frac{1}{8}$ .

$\frac{x^{2}}{16} - \frac{x}{8} = 1$

Step 2

Subtract $1$ from both sides of the equation.

$\frac{x^{2}}{16} - \frac{x}{8} - 1 = 0$

Step 3

Multiply through by the least common denominator $16$ , then simplify.

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Step 3.1

Apply the distributive property.

$16 (\frac{x^{2}}{16}) + 16 (- \frac{x}{8}) + 16 \cdot - 1 = 0$

Step 3.2

Simplify.

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Step 3.2.1

Cancel the common factor of $16$ .

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Step 3.2.1.1

Cancel the common factor.

$16 (\frac{x^{2}}{16}) + 16 (- \frac{x}{8}) + 16 \cdot - 1 = 0$

Step 3.2.1.2

Rewrite the expression.

$x^{2} + 16 (- \frac{x}{8}) + 16 \cdot - 1 = 0$

Step 3.2.2

Cancel the common factor of $8$ .

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Step 3.2.2.1

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

$x^{2} + 16 (\frac{- x}{8}) + 16 \cdot - 1 = 0$

Step 3.2.2.2

Factor $8$ out of $16$ .

$x^{2} + 8 (2) (\frac{- x}{8}) + 16 \cdot - 1 = 0$

Step 3.2.2.3

Cancel the common factor.

$x^{2} + 8 \cdot (2 (\frac{- x}{8})) + 16 \cdot - 1 = 0$

Step 3.2.2.4

Rewrite the expression.

$x^{2} + 2 (- x) + 16 \cdot - 1 = 0$

Step 3.2.3

Multiply $- 1$ by $2$ .

$x^{2} - 2 x + 16 \cdot - 1 = 0$

Step 3.2.4

Multiply $16$ by $- 1$ .

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

Step 4

Use the quadratic formula to find the solutions.

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

Step 5

Substitute the values $a = 1$ , $b = - 2$ , and $c = - 16$ into the quadratic formula and solve for $x$ .

$\frac{2 \pm \sqrt{{(- 2)}^{2} - 4 \cdot (1 \cdot - 16)}}{2 \cdot 1}$

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

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

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 1 \cdot - 16}}{2 \cdot 1}$

Step 6.1.2

Multiply $- 4 \cdot 1 \cdot - 16$ .

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Step 6.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{2 \pm \sqrt{4 - 4 \cdot - 16}}{2 \cdot 1}$

Step 6.1.2.2

Multiply $- 4$ by $- 16$ .

$x = \frac{2 \pm \sqrt{4 + 64}}{2 \cdot 1}$

Step 6.1.3

Add $4$ and $64$ .

$x = \frac{2 \pm \sqrt{68}}{2 \cdot 1}$

Step 6.1.4

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

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Step 6.1.4.1

Factor $4$ out of $68$ .

$x = \frac{2 \pm \sqrt{4 (17)}}{2 \cdot 1}$

Step 6.1.4.2

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

$x = \frac{2 \pm \sqrt{2^{2} \cdot 17}}{2 \cdot 1}$

Step 6.1.5

Pull terms out from under the radical.

$x = \frac{2 \pm 2 \sqrt{17}}{2 \cdot 1}$

Step 6.2

Multiply $2$ by $1$ .

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

Step 6.3

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

$x = 1 \pm \sqrt{17}$

Step 7

The final answer is the combination of both solutions.

$x = 1 + \sqrt{17}, 1 - \sqrt{17}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = 1 + \sqrt{17}, 1 - \sqrt{17}$

Decimal Form:

$x = 5.12310562 \dots, - 3.12310562 \dots$