Solve Using the Square Root Property -5x^2=11

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Algebra

$- 5 x^{2} = 11$

Divide each term by $- 5$ and simplify.

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Divide each term in $- 5 x^{2} = 11$ by $- 5$ .

$\frac{- 5 x^{2}}{- 5} = \frac{11}{- 5}$

Cancel the common factor of $- 5$ .

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Cancel the common factor.

$\frac{- 5 x^{2}}{- 5} = \frac{11}{- 5}$

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

$x^{2} = \frac{11}{- 5}$

Move the negative in front of the fraction.

$x^{2} = - \frac{11}{5}$

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

$x = \pm \sqrt{- \frac{11}{5}}$

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

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Simplify the right side of the equation.

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

$x = \pm \sqrt{i^{2} (\frac{11}{5})}$

Pull terms out from under the radical.

$x = \pm i \sqrt{\frac{11}{5}}$

Rewrite $\sqrt{\frac{11}{5}}$ as $\frac{\sqrt{11}}{\sqrt{5}}$ .

$x = \pm i (\frac{\sqrt{11}}{\sqrt{5}})$

Multiply $\frac{\sqrt{11}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$x = \pm i (\frac{\sqrt{11}}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}})$

Combine and simplify the denominator.

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Multiply $\frac{\sqrt{11}}{\sqrt{5}}$ and $\frac{\sqrt{5}}{\sqrt{5}}$ .

$x = \pm i (\frac{\sqrt{11} \sqrt{5}}{\sqrt{5} \sqrt{5}})$

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

$x = \pm i (\frac{\sqrt{11} \sqrt{5}}{\sqrt{5} \sqrt{5}})$

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

$x = \pm i (\frac{\sqrt{11} \sqrt{5}}{\sqrt{5} \sqrt{5}})$

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

$x = \pm i (\frac{\sqrt{11} \sqrt{5}}{{\sqrt{5}}^{1 + 1}})$

Add $1$ and $1$ .

$x = \pm i (\frac{\sqrt{11} \sqrt{5}}{{\sqrt{5}}^{2}})$

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

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

$x = \pm i (\frac{\sqrt{11} \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}})$

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

$x = \pm i (\frac{\sqrt{11} \sqrt{5}}{5^{\frac{1}{2} \cdot 2}})$

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

$x = \pm i (\frac{\sqrt{11} \sqrt{5}}{5^{\frac{2}{2}}})$

Cancel the common factor of $2$ .

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Cancel the common factor.

$x = \pm i (\frac{\sqrt{11} \sqrt{5}}{5^{\frac{2}{2}}})$

Divide $1$ by $1$ .

$x = \pm i (\frac{\sqrt{11} \sqrt{5}}{5})$

Evaluate the exponent.

$x = \pm i (\frac{\sqrt{11} \sqrt{5}}{5})$

Simplify the numerator.

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Combine using the product rule for radicals.

$x = \pm i (\frac{\sqrt{11 \cdot 5}}{5})$

Multiply $11$ by $5$ .

$x = \pm i (\frac{\sqrt{55}}{5})$

Combine $i$ and $\frac{\sqrt{55}}{5}$ .

$x = \pm \frac{i \sqrt{55}}{5}$

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

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

$x = \frac{i \sqrt{55}}{5}$

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

$x = - \frac{i \sqrt{55}}{5}$

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

$x = \frac{i \sqrt{55}}{5}, - \frac{i \sqrt{55}}{5}$

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