Pre-Algebra Examples

$9 (\frac{680}{40}) + 6 x^{2} = 89$

Step 1

Simplify each term.

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Step 1.1

Divide $680$ by $40$ .

$9 \cdot 17 + 6 x^{2} = 89$

Step 1.2

Multiply $9$ by $17$ .

$153 + 6 x^{2} = 89$

Step 2

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

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Step 2.1

Subtract $153$ from both sides of the equation.

$6 x^{2} = 89 - 153$

Step 2.2

Subtract $153$ from $89$ .

$6 x^{2} = - 64$

Step 3

Divide each term in $6 x^{2} = - 64$ by $6$ and simplify.

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Step 3.1

Divide each term in $6 x^{2} = - 64$ by $6$ .

$\frac{6 x^{2}}{6} = \frac{- 64}{6}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $6$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{6 x^{2}}{6} = \frac{- 64}{6}$

Step 3.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{- 64}{6}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Cancel the common factor of $- 64$ and $6$ .

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Step 3.3.1.1

Factor $2$ out of $- 64$ .

$x^{2} = \frac{2 (- 32)}{6}$

Step 3.3.1.2

Cancel the common factors.

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Step 3.3.1.2.1

Factor $2$ out of $6$ .

$x^{2} = \frac{2 \cdot - 32}{2 \cdot 3}$

Step 3.3.1.2.2

Cancel the common factor.

$x^{2} = \frac{2 \cdot - 32}{2 \cdot 3}$

Step 3.3.1.2.3

Rewrite the expression.

$x^{2} = \frac{- 32}{3}$

Step 3.3.2

Move the negative in front of the fraction.

$x^{2} = - \frac{32}{3}$

Step 4

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

$x = \pm \sqrt{- \frac{32}{3}}$

Step 5

Simplify $\pm \sqrt{- \frac{32}{3}}$ .

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Step 5.1

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

$x = \pm \sqrt{i^{2} \frac{32}{3}}$

Step 5.2

Pull terms out from under the radical.

$x = \pm i \sqrt{\frac{32}{3}}$

Step 5.3

Rewrite $\sqrt{\frac{32}{3}}$ as $\frac{\sqrt{32}}{\sqrt{3}}$ .

$x = \pm i \frac{\sqrt{32}}{\sqrt{3}}$

Step 5.4

Simplify the numerator.

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Step 5.4.1

Rewrite $32$ as $4^{2} \cdot 2$ .

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Step 5.4.1.1

Factor $16$ out of $32$ .

$x = \pm i \frac{\sqrt{16 (2)}}{\sqrt{3}}$

Step 5.4.1.2

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

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

Step 5.4.2

Pull terms out from under the radical.

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

Step 5.5

Multiply $\frac{4 \sqrt{2}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = \pm i (\frac{4 \sqrt{2}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})$

Step 5.6

Combine and simplify the denominator.

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Step 5.6.1

Multiply $\frac{4 \sqrt{2}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Step 5.6.2

Raise $\sqrt{3}$ to the power of $1$ .

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

Step 5.6.3

Raise $\sqrt{3}$ to the power of $1$ .

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

Step 5.6.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \pm i \frac{4 \sqrt{2} \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 5.6.5

Add $1$ and $1$ .

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

Step 5.6.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 5.6.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$x = \pm i \frac{4 \sqrt{2} \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 5.6.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 5.6.6.3

Combine $\frac{1}{2}$ and $2$ .

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

Step 5.6.6.4

Cancel the common factor of $2$ .

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Step 5.6.6.4.1

Cancel the common factor.

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

Step 5.6.6.4.2

Rewrite the expression.

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

Step 5.6.6.5

Evaluate the exponent.

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

Step 5.7

Simplify the numerator.

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Step 5.7.1

Combine using the product rule for radicals.

$x = \pm i \frac{4 \sqrt{3 \cdot 2}}{3}$

Step 5.7.2

Multiply $3$ by $2$ .

$x = \pm i \frac{4 \sqrt{6}}{3}$

Step 5.8

Combine fractions.

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Step 5.8.1

Combine $i$ and $\frac{4 \sqrt{6}}{3}$ .

$x = \pm \frac{i (4 \sqrt{6})}{3}$

Step 5.8.2

Move $4$ to the left of $i$ .

$x = \pm \frac{4 i \sqrt{6}}{3}$

Step 6

The complete solution is the result of both the positive and negative portions of the solution.

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Step 6.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \frac{4 i \sqrt{6}}{3}$

Step 6.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \frac{4 i \sqrt{6}}{3}$

Step 6.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \frac{4 i \sqrt{6}}{3}, - \frac{4 i \sqrt{6}}{3}$