Precalculus Examples

$v^{2} + 2 = 8 - 4 v^{2}$

Step 1

Move all terms containing $v$ to the left side of the equation.

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Step 1.1

Add $4 v^{2}$ to both sides of the equation.

$v^{2} + 2 + 4 v^{2} = 8$

Step 1.2

Add $v^{2}$ and $4 v^{2}$ .

$5 v^{2} + 2 = 8$

Step 2

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

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Step 2.1

Subtract $2$ from both sides of the equation.

$5 v^{2} = 8 - 2$

Step 2.2

Subtract $2$ from $8$ .

$5 v^{2} = 6$

Step 3

Divide each term in $5 v^{2} = 6$ by $5$ and simplify.

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Step 3.1

Divide each term in $5 v^{2} = 6$ by $5$ .

$\frac{5 v^{2}}{5} = \frac{6}{5}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $5$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{5 v^{2}}{5} = \frac{6}{5}$

Step 3.2.1.2

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

$v^{2} = \frac{6}{5}$

Step 4

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

$v = \pm \sqrt{\frac{6}{5}}$

Step 5

Simplify $\pm \sqrt{\frac{6}{5}}$ .

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Step 5.1

Rewrite $\sqrt{\frac{6}{5}}$ as $\frac{\sqrt{6}}{\sqrt{5}}$ .

$v = \pm \frac{\sqrt{6}}{\sqrt{5}}$

Step 5.2

Multiply $\frac{\sqrt{6}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$v = \pm \frac{\sqrt{6}}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}$

Step 5.3

Combine and simplify the denominator.

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Step 5.3.1

Multiply $\frac{\sqrt{6}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$v = \pm \frac{\sqrt{6} \sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 5.3.2

Raise $\sqrt{5}$ to the power of $1$ .

$v = \pm \frac{\sqrt{6} \sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}}$

Step 5.3.3

Raise $\sqrt{5}$ to the power of $1$ .

$v = \pm \frac{\sqrt{6} \sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}}$

Step 5.3.4

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

$v = \pm \frac{\sqrt{6} \sqrt{5}}{{\sqrt{5}}^{1 + 1}}$

Step 5.3.5

Add $1$ and $1$ .

$v = \pm \frac{\sqrt{6} \sqrt{5}}{{\sqrt{5}}^{2}}$

Step 5.3.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 5.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$v = \pm \frac{\sqrt{6} \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}$

Step 5.3.6.2

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

$v = \pm \frac{\sqrt{6} \sqrt{5}}{5^{\frac{1}{2} \cdot 2}}$

Step 5.3.6.3

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

$v = \pm \frac{\sqrt{6} \sqrt{5}}{5^{\frac{2}{2}}}$

Step 5.3.6.4

Cancel the common factor of $2$ .

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Step 5.3.6.4.1

Cancel the common factor.

$v = \pm \frac{\sqrt{6} \sqrt{5}}{5^{\frac{2}{2}}}$

Step 5.3.6.4.2

Rewrite the expression.

$v = \pm \frac{\sqrt{6} \sqrt{5}}{5^{1}}$

Step 5.3.6.5

Evaluate the exponent.

$v = \pm \frac{\sqrt{6} \sqrt{5}}{5}$

Step 5.4

Simplify the numerator.

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Step 5.4.1

Combine using the product rule for radicals.

$v = \pm \frac{\sqrt{6 \cdot 5}}{5}$

Step 5.4.2

Multiply $6$ by $5$ .

$v = \pm \frac{\sqrt{30}}{5}$

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.

$v = \frac{\sqrt{30}}{5}$

Step 6.2

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

$v = - \frac{\sqrt{30}}{5}$

Step 6.3

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

$v = \frac{\sqrt{30}}{5}, - \frac{\sqrt{30}}{5}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$v = \frac{\sqrt{30}}{5}, - \frac{\sqrt{30}}{5}$

Decimal Form:

$v = 1.09544511 \dots, - 1.09544511 \dots$