Find the Roots (Zeros) 7x^2+1=0

$7 x^{2} + 1 = 0$

Step 1

Subtract $1$ from both sides of the equation.

$7 x^{2} = - 1$

Step 2

Divide each term in $7 x^{2} = - 1$ by $7$ and simplify.

Divide each term in $7 x^{2} = - 1$ by $7$ .

$\frac{7 x^{2}}{7} = \frac{- 1}{7}$

Simplify the left side.

Cancel the common factor of $7$ .

Cancel the common factor.

$\frac{7 x^{2}}{7} = \frac{- 1}{7}$

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

$x^{2} = \frac{- 1}{7}$

Simplify the right side.

Move the negative in front of the fraction.

$x^{2} = - \frac{1}{7}$

Step 3

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

$x = \pm \sqrt{- \frac{1}{7}}$

Step 4

Simplify $\pm \sqrt{- \frac{1}{7}}$ .

Rewrite $- \frac{1}{7}$ as $i^{2} \frac{1^{2}}{7}$ .

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

$x = \pm \sqrt{i^{2} \frac{1}{7}}$

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

$x = \pm \sqrt{i^{2} \frac{1^{2}}{7}}$

Pull terms out from under the radical.

$x = \pm i \sqrt{\frac{1^{2}}{7}}$

One to any power is one.

$x = \pm i \sqrt{\frac{1}{7}}$

Rewrite $\sqrt{\frac{1}{7}}$ as $\frac{\sqrt{1}}{\sqrt{7}}$ .

$x = \pm i \frac{\sqrt{1}}{\sqrt{7}}$

Any root of $1$ is $1$ .

$x = \pm i \frac{1}{\sqrt{7}}$

Multiply $\frac{1}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$x = \pm i (\frac{1}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}})$

Combine and simplify the denominator.

Multiply $\frac{1}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$x = \pm i \frac{\sqrt{7}}{\sqrt{7} \sqrt{7}}$

Raise $\sqrt{7}$ to the power of $1$ .

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

Raise $\sqrt{7}$ to the power of $1$ .

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

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

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

Add $1$ and $1$ .

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

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

$x = \pm i \frac{\sqrt{7}}{7}$

Combine $i$ and $\frac{\sqrt{7}}{7}$ .

$x = \pm \frac{i \sqrt{7}}{7}$

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.

$x = \frac{i \sqrt{7}}{7}$

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

$x = - \frac{i \sqrt{7}}{7}$

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

$x = \frac{i \sqrt{7}}{7}, - \frac{i \sqrt{7}}{7}$

Step 6

Find the Roots (Zeros) 7x^2+1=0

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