Precalculus Examples

$\frac{3 x^{2}}{2} + 8 = \frac{26}{3}$

Step 1

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

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Step 1.1

Subtract $8$ from both sides of the equation.

$\frac{3 x^{2}}{2} = \frac{26}{3} - 8$

Step 1.2

To write $- 8$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{3 x^{2}}{2} = \frac{26}{3} - 8 \cdot \frac{3}{3}$

Step 1.3

Combine $- 8$ and $\frac{3}{3}$ .

$\frac{3 x^{2}}{2} = \frac{26}{3} + \frac{- 8 \cdot 3}{3}$

Step 1.4

Combine the numerators over the common denominator.

$\frac{3 x^{2}}{2} = \frac{26 - 8 \cdot 3}{3}$

Step 1.5

Simplify the numerator.

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Step 1.5.1

Multiply $- 8$ by $3$ .

$\frac{3 x^{2}}{2} = \frac{26 - 24}{3}$

Step 1.5.2

Subtract $24$ from $26$ .

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

Step 2

Multiply both sides of the equation by $\frac{2}{3}$ .

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

Step 3

Simplify both sides of the equation.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Simplify $\frac{2}{3} \cdot \frac{3 x^{2}}{2}$ .

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Step 3.1.1.1

Combine.

$\frac{2 (3 x^{2})}{3 \cdot 2} = \frac{2}{3} \cdot \frac{2}{3}$

Step 3.1.1.2

Cancel the common factor of $2$ .

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Step 3.1.1.2.1

Cancel the common factor.

$\frac{2 (3 x^{2})}{3 \cdot 2} = \frac{2}{3} \cdot \frac{2}{3}$

Step 3.1.1.2.2

Rewrite the expression.

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

Step 3.1.1.3

Cancel the common factor of $3$ .

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Step 3.1.1.3.1

Cancel the common factor.

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

Step 3.1.1.3.2

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

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

Step 3.2

Simplify the right side.

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Step 3.2.1

Multiply $\frac{2}{3} \cdot \frac{2}{3}$ .

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Step 3.2.1.1

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

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

Step 3.2.1.2

Multiply $2$ by $2$ .

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

Step 3.2.1.3

Multiply $3$ by $3$ .

$x^{2} = \frac{4}{9}$

Step 4

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

$x = \pm \sqrt{\frac{4}{9}}$

Step 5

Simplify $\pm \sqrt{\frac{4}{9}}$ .

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Step 5.1

Rewrite $\sqrt{\frac{4}{9}}$ as $\frac{\sqrt{4}}{\sqrt{9}}$ .

$x = \pm \frac{\sqrt{4}}{\sqrt{9}}$

Step 5.2

Simplify the numerator.

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Step 5.2.1

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

$x = \pm \frac{\sqrt{2^{2}}}{\sqrt{9}}$

Step 5.2.2

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

$x = \pm \frac{2}{\sqrt{9}}$

Step 5.3

Simplify the denominator.

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Step 5.3.1

Rewrite $9$ as $3^{2}$ .

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

Step 5.3.2

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

$x = \pm \frac{2}{3}$

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.

$x = \frac{2}{3}$

Step 6.2

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

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

Step 6.3

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

$x = \frac{2}{3}, - \frac{2}{3}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \frac{2}{3}, - \frac{2}{3}$

Decimal Form:

$x = 0.\overline{6}, - 0.\overline{6}$