Pre-Algebra Examples

$\frac{1}{16} x^{2} + \frac{1}{18} x = \frac{1}{2}$

Step 1

Simplify each term.

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Step 1.1

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

$\frac{x^{2}}{16} + \frac{1}{18} x = \frac{1}{2}$

Step 1.2

Combine $\frac{1}{18}$ and $x$ .

$\frac{x^{2}}{16} + \frac{x}{18} = \frac{1}{2}$

Step 2

Subtract $\frac{1}{2}$ from both sides of the equation.

$\frac{x^{2}}{16} + \frac{x}{18} - \frac{1}{2} = 0$

Step 3

Multiply through by the least common denominator $144$ , then simplify.

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Step 3.1

Apply the distributive property.

$144 (\frac{x^{2}}{16}) + 144 (\frac{x}{18}) + 144 (- \frac{1}{2}) = 0$

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

Factor $16$ out of $144$ .

$16 (9) (\frac{x^{2}}{16}) + 144 (\frac{x}{18}) + 144 (- \frac{1}{2}) = 0$

Step 3.2.1.2

Cancel the common factor.

$16 \cdot (9 (\frac{x^{2}}{16})) + 144 (\frac{x}{18}) + 144 (- \frac{1}{2}) = 0$

Step 3.2.1.3

Rewrite the expression.

$9 x^{2} + 144 (\frac{x}{18}) + 144 (- \frac{1}{2}) = 0$

Step 3.2.2

Cancel the common factor of $18$ .

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Step 3.2.2.1

Factor $18$ out of $144$ .

$9 x^{2} + 18 (8) (\frac{x}{18}) + 144 (- \frac{1}{2}) = 0$

Step 3.2.2.2

Cancel the common factor.

$9 x^{2} + 18 \cdot (8 (\frac{x}{18})) + 144 (- \frac{1}{2}) = 0$

Step 3.2.2.3

Rewrite the expression.

$9 x^{2} + 8 x + 144 (- \frac{1}{2}) = 0$

Step 3.2.3

Cancel the common factor of $2$ .

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Step 3.2.3.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$9 x^{2} + 8 x + 144 (\frac{- 1}{2}) = 0$

Step 3.2.3.2

Factor $2$ out of $144$ .

$9 x^{2} + 8 x + 2 (72) (\frac{- 1}{2}) = 0$

Step 3.2.3.3

Cancel the common factor.

$9 x^{2} + 8 x + 2 \cdot (72 (\frac{- 1}{2})) = 0$

Step 3.2.3.4

Rewrite the expression.

$9 x^{2} + 8 x + 72 \cdot - 1 = 0$

Step 3.2.4

Multiply $72$ by $- 1$ .

$9 x^{2} + 8 x - 72 = 0$

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 $a = 9$ , $b = 8$ , and $c = - 72$ into the quadratic formula and solve for $x$ .

$\frac{- 8 \pm \sqrt{8^{2} - 4 \cdot (9 \cdot - 72)}}{2 \cdot 9}$

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise $8$ to the power of $2$ .

$x = \frac{- 8 \pm \sqrt{64 - 4 \cdot 9 \cdot - 72}}{2 \cdot 9}$

Step 6.1.2

Multiply $- 4 \cdot 9 \cdot - 72$ .

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Step 6.1.2.1

Multiply $- 4$ by $9$ .

$x = \frac{- 8 \pm \sqrt{64 - 36 \cdot - 72}}{2 \cdot 9}$

Step 6.1.2.2

Multiply $- 36$ by $- 72$ .

$x = \frac{- 8 \pm \sqrt{64 + 2592}}{2 \cdot 9}$

Step 6.1.3

Add $64$ and $2592$ .

$x = \frac{- 8 \pm \sqrt{2656}}{2 \cdot 9}$

Step 6.1.4

Rewrite $2656$ as $4^{2} \cdot 166$ .

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Step 6.1.4.1

Factor $16$ out of $2656$ .

$x = \frac{- 8 \pm \sqrt{16 (166)}}{2 \cdot 9}$

Step 6.1.4.2

Rewrite $16$ as $4^{2}$ .

$x = \frac{- 8 \pm \sqrt{4^{2} \cdot 166}}{2 \cdot 9}$

Step 6.1.5

Pull terms out from under the radical.

$x = \frac{- 8 \pm 4 \sqrt{166}}{2 \cdot 9}$

Step 6.2

Multiply $2$ by $9$ .

$x = \frac{- 8 \pm 4 \sqrt{166}}{18}$

Step 6.3

Simplify $\frac{- 8 \pm 4 \sqrt{166}}{18}$ .

$x = \frac{- 4 \pm 2 \sqrt{166}}{9}$

Step 7

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Raise $8$ to the power of $2$ .

$x = \frac{- 8 \pm \sqrt{64 - 4 \cdot 9 \cdot - 72}}{2 \cdot 9}$

Step 7.1.2

Multiply $- 4 \cdot 9 \cdot - 72$ .

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Step 7.1.2.1

Multiply $- 4$ by $9$ .

$x = \frac{- 8 \pm \sqrt{64 - 36 \cdot - 72}}{2 \cdot 9}$

Step 7.1.2.2

Multiply $- 36$ by $- 72$ .

$x = \frac{- 8 \pm \sqrt{64 + 2592}}{2 \cdot 9}$

Step 7.1.3

Add $64$ and $2592$ .

$x = \frac{- 8 \pm \sqrt{2656}}{2 \cdot 9}$

Step 7.1.4

Rewrite $2656$ as $4^{2} \cdot 166$ .

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Step 7.1.4.1

Factor $16$ out of $2656$ .

$x = \frac{- 8 \pm \sqrt{16 (166)}}{2 \cdot 9}$

Step 7.1.4.2

Rewrite $16$ as $4^{2}$ .

$x = \frac{- 8 \pm \sqrt{4^{2} \cdot 166}}{2 \cdot 9}$

Step 7.1.5

Pull terms out from under the radical.

$x = \frac{- 8 \pm 4 \sqrt{166}}{2 \cdot 9}$

Step 7.2

Multiply $2$ by $9$ .

$x = \frac{- 8 \pm 4 \sqrt{166}}{18}$

Step 7.3

Simplify $\frac{- 8 \pm 4 \sqrt{166}}{18}$ .

$x = \frac{- 4 \pm 2 \sqrt{166}}{9}$

Step 7.4

Change the $\pm$ to $+$ .

$x = \frac{- 4 + 2 \sqrt{166}}{9}$

Step 7.5

Rewrite $- 4$ as $- 1 (4)$ .

$x = \frac{- 1 \cdot 4 + 2 \sqrt{166}}{9}$

Step 7.6

Factor $- 1$ out of $2 \sqrt{166}$ .

$x = \frac{- 1 \cdot 4 - (- 2 \sqrt{166})}{9}$

Step 7.7

Factor $- 1$ out of $- 1 (4) - (- 2 \sqrt{166})$ .

$x = \frac{- 1 (4 - 2 \sqrt{166})}{9}$

Step 7.8

Move the negative in front of the fraction.

$x = - \frac{4 - 2 \sqrt{166}}{9}$

Step 8

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Raise $8$ to the power of $2$ .

$x = \frac{- 8 \pm \sqrt{64 - 4 \cdot 9 \cdot - 72}}{2 \cdot 9}$

Step 8.1.2

Multiply $- 4 \cdot 9 \cdot - 72$ .

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Step 8.1.2.1

Multiply $- 4$ by $9$ .

$x = \frac{- 8 \pm \sqrt{64 - 36 \cdot - 72}}{2 \cdot 9}$

Step 8.1.2.2

Multiply $- 36$ by $- 72$ .

$x = \frac{- 8 \pm \sqrt{64 + 2592}}{2 \cdot 9}$

Step 8.1.3

Add $64$ and $2592$ .

$x = \frac{- 8 \pm \sqrt{2656}}{2 \cdot 9}$

Step 8.1.4

Rewrite $2656$ as $4^{2} \cdot 166$ .

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Step 8.1.4.1

Factor $16$ out of $2656$ .

$x = \frac{- 8 \pm \sqrt{16 (166)}}{2 \cdot 9}$

Step 8.1.4.2

Rewrite $16$ as $4^{2}$ .

$x = \frac{- 8 \pm \sqrt{4^{2} \cdot 166}}{2 \cdot 9}$

Step 8.1.5

Pull terms out from under the radical.

$x = \frac{- 8 \pm 4 \sqrt{166}}{2 \cdot 9}$

Step 8.2

Multiply $2$ by $9$ .

$x = \frac{- 8 \pm 4 \sqrt{166}}{18}$

Step 8.3

Simplify $\frac{- 8 \pm 4 \sqrt{166}}{18}$ .

$x = \frac{- 4 \pm 2 \sqrt{166}}{9}$

Step 8.4

Change the $\pm$ to $-$ .

$x = \frac{- 4 - 2 \sqrt{166}}{9}$

Step 8.5

Rewrite $- 4$ as $- 1 (4)$ .

$x = \frac{- 1 \cdot 4 - 2 \sqrt{166}}{9}$

Step 8.6

Factor $- 1$ out of $- 2 \sqrt{166}$ .

$x = \frac{- 1 \cdot 4 - (2 \sqrt{166})}{9}$

Step 8.7

Factor $- 1$ out of $- 1 (4) - (2 \sqrt{166})$ .

$x = \frac{- 1 (4 + 2 \sqrt{166})}{9}$

Step 8.8

Move the negative in front of the fraction.

$x = - \frac{4 + 2 \sqrt{166}}{9}$

Step 9

The final answer is the combination of both solutions.

$x = - \frac{4 - 2 \sqrt{166}}{9}, - \frac{4 + 2 \sqrt{166}}{9}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{4 - 2 \sqrt{166}}{9}, - \frac{4 + 2 \sqrt{166}}{9}$

Decimal Form:

$x = 2.41868860 \dots, - 3.30757749 \dots$