Algebra Examples

$\sqrt{3 x - 11 + 5} = x$

Step 1

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

${\sqrt{3 x - 11 + 5}}^{2} = x^{2}$

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 $\sqrt{3 x - 11 + 5}$ as ${(3 x - 11 + 5)}^{\frac{1}{2}}$ .

${({(3 x - 11 + 5)}^{\frac{1}{2}})}^{2} = x^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(3 x - 11 + 5)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

Multiply the exponents in ${({(3 x - 11 + 5)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1.1

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

${(3 x - 11 + 5)}^{\frac{1}{2} \cdot 2} = x^{2}$

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.

${(3 x - 11 + 5)}^{\frac{1}{2} \cdot 2} = x^{2}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(3 x - 11 + 5)}^{1} = x^{2}$

Step 2.2.1.2

Add $- 11$ and $5$ .

${(3 x - 6)}^{1} = x^{2}$

Step 2.2.1.3

Simplify.

$3 x - 6 = x^{2}$

Step 3

Solve for $x$ .

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Step 3.1

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

$3 x - 6 - x^{2} = 0$

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 $a = - 1$ , $b = 3$ , and $c = - 6$ into the quadratic formula and solve for $x$ .

$\frac{- 3 \pm \sqrt{3^{2} - 4 \cdot (- 1 \cdot - 6)}}{2 \cdot - 1}$

Step 3.4

Simplify.

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Step 3.4.1

Simplify the numerator.

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Step 3.4.1.1

Raise $3$ to the power of $2$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot - 1 \cdot - 6}}{2 \cdot - 1}$

Step 3.4.1.2

Multiply $- 4 \cdot - 1 \cdot - 6$ .

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Step 3.4.1.2.1

Multiply $- 4$ by $- 1$ .

$x = \frac{- 3 \pm \sqrt{9 + 4 \cdot - 6}}{2 \cdot - 1}$

Step 3.4.1.2.2

Multiply $4$ by $- 6$ .

$x = \frac{- 3 \pm \sqrt{9 - 24}}{2 \cdot - 1}$

Step 3.4.1.3

Subtract $24$ from $9$ .

$x = \frac{- 3 \pm \sqrt{- 15}}{2 \cdot - 1}$

Step 3.4.1.4

Rewrite $- 15$ as $- 1 (15)$ .

$x = \frac{- 3 \pm \sqrt{- 1 \cdot 15}}{2 \cdot - 1}$

Step 3.4.1.5

Rewrite $\sqrt{- 1 (15)}$ as $\sqrt{- 1} \cdot \sqrt{15}$ .

$x = \frac{- 3 \pm \sqrt{- 1} \cdot \sqrt{15}}{2 \cdot - 1}$

Step 3.4.1.6

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

$x = \frac{- 3 \pm i \sqrt{15}}{2 \cdot - 1}$

Step 3.4.2

Multiply $2$ by $- 1$ .

$x = \frac{- 3 \pm i \sqrt{15}}{- 2}$

Step 3.4.3

Simplify $\frac{- 3 \pm i \sqrt{15}}{- 2}$ .

$x = \frac{3 \pm i \sqrt{15}}{2}$

Step 3.5

The final answer is the combination of both solutions.

$x = \frac{3 + i \sqrt{15}}{2}, \frac{3 - i \sqrt{15}}{2}$