Solve for v 5v^2+2=-5

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

Step 1

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

Subtract $2$ from both sides of the equation.

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

Subtract $2$ from $- 5$ .

$5 v^{2} = - 7$

Step 2

Divide each term in $5 v^{2} = - 7$ by $5$ and simplify.

Divide each term in $5 v^{2} = - 7$ by $5$ .

$\frac{5 v^{2}}{5} = \frac{- 7}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 v^{2}}{5} = \frac{- 7}{5}$

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

$v^{2} = \frac{- 7}{5}$

Simplify the right side.

Move the negative in front of the fraction.

$v^{2} = - \frac{7}{5}$

Step 3

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

$v = \pm \sqrt{- \frac{7}{5}}$

Step 4

Simplify $\pm \sqrt{- \frac{7}{5}}$ .

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

$v = \pm \sqrt{i^{2} \frac{7}{5}}$

Pull terms out from under the radical.

$v = \pm i \sqrt{\frac{7}{5}}$

Rewrite $\sqrt{\frac{7}{5}}$ as $\frac{\sqrt{7}}{\sqrt{5}}$ .

$v = \pm i \frac{\sqrt{7}}{\sqrt{5}}$

Multiply $\frac{\sqrt{7}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$v = \pm i (\frac{\sqrt{7}}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}})$

Combine and simplify the denominator.

Multiply $\frac{\sqrt{7}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$v = \pm i \frac{\sqrt{7} \sqrt{5}}{\sqrt{5} \sqrt{5}}$

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

$v = \pm i \frac{\sqrt{7} \sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}}$

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

$v = \pm i \frac{\sqrt{7} \sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}}$

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

$v = \pm i \frac{\sqrt{7} \sqrt{5}}{{\sqrt{5}}^{1 + 1}}$

Add $1$ and $1$ .

$v = \pm i \frac{\sqrt{7} \sqrt{5}}{{\sqrt{5}}^{2}}$

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

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

$v = \pm i \frac{\sqrt{7} \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}$

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

$v = \pm i \frac{\sqrt{7} \sqrt{5}}{5^{\frac{1}{2} \cdot 2}}$

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

$v = \pm i \frac{\sqrt{7} \sqrt{5}}{5^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$v = \pm i \frac{\sqrt{7} \sqrt{5}}{5^{\frac{2}{2}}}$

Rewrite the expression.

$v = \pm i \frac{\sqrt{7} \sqrt{5}}{5^{1}}$

Evaluate the exponent.

$v = \pm i \frac{\sqrt{7} \sqrt{5}}{5}$

Simplify the numerator.

Combine using the product rule for radicals.

$v = \pm i \frac{\sqrt{7 \cdot 5}}{5}$

Multiply $7$ by $5$ .

$v = \pm i \frac{\sqrt{35}}{5}$

Combine $i$ and $\frac{\sqrt{35}}{5}$ .

$v = \pm \frac{i \sqrt{35}}{5}$

Step 5

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

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

$v = \frac{i \sqrt{35}}{5}$

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

$v = - \frac{i \sqrt{35}}{5}$

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

$v = \frac{i \sqrt{35}}{5}, - \frac{i \sqrt{35}}{5}$

Solve for v 5v^2+2=-5

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