Solve Using the Square Root Property 11x^2+7=0

$11 x^{2} + 7 = 0$

Step 1

Subtract $7$ from both sides of the equation.

$11 x^{2} = - 7$

Step 2

Divide each term in $11 x^{2} = - 7$ by $11$ and simplify.

Divide each term in $11 x^{2} = - 7$ by $11$ .

$\frac{11 x^{2}}{11} = \frac{- 7}{11}$

Simplify the left side.

Cancel the common factor of $11$ .

Cancel the common factor.

$\frac{11 x^{2}}{11} = \frac{- 7}{11}$

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

$x^{2} = \frac{- 7}{11}$

Simplify the right side.

Move the negative in front of the fraction.

$x^{2} = - \frac{7}{11}$

Step 3

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

$x = \pm \sqrt{- \frac{7}{11}}$

Step 4

Simplify $\pm \sqrt{- \frac{7}{11}}$ .

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

$x = \pm \sqrt{i^{2} \frac{7}{11}}$

Pull terms out from under the radical.

$x = \pm i \sqrt{\frac{7}{11}}$

Rewrite $\sqrt{\frac{7}{11}}$ as $\frac{\sqrt{7}}{\sqrt{11}}$ .

$x = \pm i \frac{\sqrt{7}}{\sqrt{11}}$

Multiply $\frac{\sqrt{7}}{\sqrt{11}}$ by $\frac{\sqrt{11}}{\sqrt{11}}$ .

$x = \pm i (\frac{\sqrt{7}}{\sqrt{11}} \cdot \frac{\sqrt{11}}{\sqrt{11}})$

Combine and simplify the denominator.

Multiply $\frac{\sqrt{7}}{\sqrt{11}}$ by $\frac{\sqrt{11}}{\sqrt{11}}$ .

$x = \pm i \frac{\sqrt{7} \sqrt{11}}{\sqrt{11} \sqrt{11}}$

Raise $\sqrt{11}$ to the power of $1$ .

$x = \pm i \frac{\sqrt{7} \sqrt{11}}{{\sqrt{11}}^{1} \sqrt{11}}$

Raise $\sqrt{11}$ to the power of $1$ .

$x = \pm i \frac{\sqrt{7} \sqrt{11}}{{\sqrt{11}}^{1} {\sqrt{11}}^{1}}$

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

$x = \pm i \frac{\sqrt{7} \sqrt{11}}{{\sqrt{11}}^{1 + 1}}$

Add $1$ and $1$ .

$x = \pm i \frac{\sqrt{7} \sqrt{11}}{{\sqrt{11}}^{2}}$

Rewrite ${\sqrt{11}}^{2}$ as $11$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{11}$ as $11^{\frac{1}{2}}$ .

$x = \pm i \frac{\sqrt{7} \sqrt{11}}{{(11^{\frac{1}{2}})}^{2}}$

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

$x = \pm i \frac{\sqrt{7} \sqrt{11}}{11^{\frac{1}{2} \cdot 2}}$

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

$x = \pm i \frac{\sqrt{7} \sqrt{11}}{11^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x = \pm i \frac{\sqrt{7} \sqrt{11}}{11^{\frac{2}{2}}}$

Rewrite the expression.

$x = \pm i \frac{\sqrt{7} \sqrt{11}}{11^{1}}$

Evaluate the exponent.

$x = \pm i \frac{\sqrt{7} \sqrt{11}}{11}$

Simplify the numerator.

Combine using the product rule for radicals.

$x = \pm i \frac{\sqrt{7 \cdot 11}}{11}$

Multiply $7$ by $11$ .

$x = \pm i \frac{\sqrt{77}}{11}$

Combine $i$ and $\frac{\sqrt{77}}{11}$ .

$x = \pm \frac{i \sqrt{77}}{11}$

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.

$x = \frac{i \sqrt{77}}{11}$

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

$x = - \frac{i \sqrt{77}}{11}$

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

$x = \frac{i \sqrt{77}}{11}, - \frac{i \sqrt{77}}{11}$

Solve Using the Square Root Property 11x^2+7=0

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