Algebra Examples

$\sqrt{x - 2} = x$

Step 1

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

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

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 $\sqrt{x - 2}$ as ${(x - 2)}^{\frac{1}{2}}$ .

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

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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Step 2.2.1.1

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

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Step 2.2.1.1.1

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

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

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.

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

Step 2.2.1.1.2.2

Rewrite the expression.

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

Step 2.2.1.2

Simplify.

$x - 2 = x^{2}$

Step 3

Solve for $x$ .

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Step 3.1

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

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

Step 3.2

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 3.3

Substitute the values $a = - 1$ , $b = 1$ , and $c = - 2$ into the quadratic formula and solve for $x$ .

$\frac{- 1 \pm \sqrt{1^{2} - 4 \cdot (- 1 \cdot - 2)}}{2 \cdot - 1}$

Step 3.4

Simplify.

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Step 3.4.1

Simplify the numerator.

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Step 3.4.1.1

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot - 1 \cdot - 2}}{2 \cdot - 1}$

Step 3.4.1.2

Multiply $- 4 \cdot - 1 \cdot - 2$ .

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Step 3.4.1.2.1

Multiply $- 4$ by $- 1$ .

$x = \frac{- 1 \pm \sqrt{1 + 4 \cdot - 2}}{2 \cdot - 1}$

Step 3.4.1.2.2

Multiply $4$ by $- 2$ .

$x = \frac{- 1 \pm \sqrt{1 - 8}}{2 \cdot - 1}$

Step 3.4.1.3

Subtract $8$ from $1$ .

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

Step 3.4.1.4

Rewrite $- 7$ as $- 1 (7)$ .

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

Step 3.4.1.5

Rewrite $\sqrt{- 1 (7)}$ as $\sqrt{- 1} \cdot \sqrt{7}$ .

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

Step 3.4.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

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

Step 3.4.2

Multiply $2$ by $- 1$ .

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

Step 3.4.3

Simplify $\frac{- 1 \pm i \sqrt{7}}{- 2}$ .

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

Step 3.5

The final answer is the combination of both solutions.

$x = \frac{1 + i \sqrt{7}}{2}, \frac{1 - i \sqrt{7}}{2}$