Algebra Examples

$\sqrt{2 x^{2} - 121} = x$

Step 1

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{2 x^{2} - 121}}^{2} = x^{2}$

Step 2

Simplify each side of the equation.

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 x^{2} - 121}$ as ${(2 x^{2} - 121)}^{\frac{1}{2}}$ .

${({(2 x^{2} - 121)}^{\frac{1}{2}})}^{2} = x^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(2 x^{2} - 121)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

Multiply the exponents in ${({(2 x^{2} - 121)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1.1

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

${(2 x^{2} - 121)}^{\frac{1}{2} \cdot 2} = x^{2}$

Step 2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.1.2.1

Cancel the common factor.

${(2 x^{2} - 121)}^{\frac{1}{2} \cdot 2} = x^{2}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(2 x^{2} - 121)}^{1} = x^{2}$

Step 2.2.1.2

Simplify.

$2 x^{2} - 121 = x^{2}$

Step 3

Solve for $x$ .

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Step 3.1

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

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Step 3.1.1

Subtract $x^{2}$ from both sides of the equation.

$2 x^{2} - 121 - x^{2} = 0$

Step 3.1.2

Subtract $x^{2}$ from $2 x^{2}$ .

$x^{2} - 121 = 0$

Step 3.2

Add $121$ to both sides of the equation.

$x^{2} = 121$

Step 3.3

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

$x = \pm \sqrt{121}$

Step 3.4

Simplify $\pm \sqrt{121}$ .

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Step 3.4.1

Rewrite $121$ as $11^{2}$ .

$x = \pm \sqrt{11^{2}}$

Step 3.4.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 11$

Step 3.5

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

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Step 3.5.1

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

$x = 11$

Step 3.5.2

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

$x = - 11$

Step 3.5.3

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

$x = 11, - 11$

Step 4

Exclude the solutions that do not make $\sqrt{2 x^{2} - 121} = x$ true.

$x = 11$