Algebra Examples

$\sqrt{13 x - 10} = 3 x$

Step 1

Subtract $3 x$ from both sides of the equation.

$\sqrt{13 x - 10} - 3 x = 0$

Step 2

Add $3 x$ to both sides of the equation.

$\sqrt{13 x - 10} = 3 x$

Step 3

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

${\sqrt{13 x - 10}}^{2} = {(3 x)}^{2}$

Step 4

Simplify each side of the equation.

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Step 4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{13 x - 10}$ as ${(13 x - 10)}^{\frac{1}{2}}$ .

${({(13 x - 10)}^{\frac{1}{2}})}^{2} = {(3 x)}^{2}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify ${({(13 x - 10)}^{\frac{1}{2}})}^{2}$ .

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Step 4.2.1.1

Multiply the exponents in ${({(13 x - 10)}^{\frac{1}{2}})}^{2}$ .

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Step 4.2.1.1.1

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

${(13 x - 10)}^{\frac{1}{2} \cdot 2} = {(3 x)}^{2}$

Step 4.2.1.1.2

Cancel the common factor of $2$ .

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Step 4.2.1.1.2.1

Cancel the common factor.

${(13 x - 10)}^{\frac{1}{2} \cdot 2} = {(3 x)}^{2}$

Step 4.2.1.1.2.2

Rewrite the expression.

${(13 x - 10)}^{1} = {(3 x)}^{2}$

Step 4.2.1.2

Simplify.

$13 x - 10 = {(3 x)}^{2}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Simplify ${(3 x)}^{2}$ .

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Step 4.3.1.1

Apply the product rule to $3 x$ .

$13 x - 10 = 3^{2} x^{2}$

Step 4.3.1.2

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

$13 x - 10 = 9 x^{2}$

Step 5

Solve for $x$ .

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Step 5.1

Subtract $9 x^{2}$ from both sides of the equation.

$13 x - 10 - 9 x^{2} = 0$

Step 5.2

Use the quadratic formula to find the solutions.

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

Step 5.3

Substitute the values $a = - 9$ , $b = 13$ , and $c = - 10$ into the quadratic formula and solve for $x$ .

$\frac{- 13 \pm \sqrt{13^{2} - 4 \cdot (- 9 \cdot - 10)}}{2 \cdot - 9}$

Step 5.4

Simplify.

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Step 5.4.1

Simplify the numerator.

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Step 5.4.1.1

Raise $13$ to the power of $2$ .

$x = \frac{- 13 \pm \sqrt{169 - 4 \cdot - 9 \cdot - 10}}{2 \cdot - 9}$

Step 5.4.1.2

Multiply $- 4 \cdot - 9 \cdot - 10$ .

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Step 5.4.1.2.1

Multiply $- 4$ by $- 9$ .

$x = \frac{- 13 \pm \sqrt{169 + 36 \cdot - 10}}{2 \cdot - 9}$

Step 5.4.1.2.2

Multiply $36$ by $- 10$ .

$x = \frac{- 13 \pm \sqrt{169 - 360}}{2 \cdot - 9}$

Step 5.4.1.3

Subtract $360$ from $169$ .

$x = \frac{- 13 \pm \sqrt{- 191}}{2 \cdot - 9}$

Step 5.4.1.4

Rewrite $- 191$ as $- 1 (191)$ .

$x = \frac{- 13 \pm \sqrt{- 1 \cdot 191}}{2 \cdot - 9}$

Step 5.4.1.5

Rewrite $\sqrt{- 1 (191)}$ as $\sqrt{- 1} \cdot \sqrt{191}$ .

$x = \frac{- 13 \pm \sqrt{- 1} \cdot \sqrt{191}}{2 \cdot - 9}$

Step 5.4.1.6

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

$x = \frac{- 13 \pm i \sqrt{191}}{2 \cdot - 9}$

Step 5.4.2

Multiply $2$ by $- 9$ .

$x = \frac{- 13 \pm i \sqrt{191}}{- 18}$

Step 5.4.3

Simplify $\frac{- 13 \pm i \sqrt{191}}{- 18}$ .

$x = \frac{13 \pm i \sqrt{191}}{18}$

Step 5.5

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 5.5.1

Simplify the numerator.

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Step 5.5.1.1

Raise $13$ to the power of $2$ .

$x = \frac{- 13 \pm \sqrt{169 - 4 \cdot - 9 \cdot - 10}}{2 \cdot - 9}$

Step 5.5.1.2

Multiply $- 4 \cdot - 9 \cdot - 10$ .

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Step 5.5.1.2.1

Multiply $- 4$ by $- 9$ .

$x = \frac{- 13 \pm \sqrt{169 + 36 \cdot - 10}}{2 \cdot - 9}$

Step 5.5.1.2.2

Multiply $36$ by $- 10$ .

$x = \frac{- 13 \pm \sqrt{169 - 360}}{2 \cdot - 9}$

Step 5.5.1.3

Subtract $360$ from $169$ .

$x = \frac{- 13 \pm \sqrt{- 191}}{2 \cdot - 9}$

Step 5.5.1.4

Rewrite $- 191$ as $- 1 (191)$ .

$x = \frac{- 13 \pm \sqrt{- 1 \cdot 191}}{2 \cdot - 9}$

Step 5.5.1.5

Rewrite $\sqrt{- 1 (191)}$ as $\sqrt{- 1} \cdot \sqrt{191}$ .

$x = \frac{- 13 \pm \sqrt{- 1} \cdot \sqrt{191}}{2 \cdot - 9}$

Step 5.5.1.6

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

$x = \frac{- 13 \pm i \sqrt{191}}{2 \cdot - 9}$

Step 5.5.2

Multiply $2$ by $- 9$ .

$x = \frac{- 13 \pm i \sqrt{191}}{- 18}$

Step 5.5.3

Simplify $\frac{- 13 \pm i \sqrt{191}}{- 18}$ .

$x = \frac{13 \pm i \sqrt{191}}{18}$

Step 5.5.4

Change the $\pm$ to $+$ .

$x = \frac{13 + i \sqrt{191}}{18}$

Step 5.6

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 5.6.1

Simplify the numerator.

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Step 5.6.1.1

Raise $13$ to the power of $2$ .

$x = \frac{- 13 \pm \sqrt{169 - 4 \cdot - 9 \cdot - 10}}{2 \cdot - 9}$

Step 5.6.1.2

Multiply $- 4 \cdot - 9 \cdot - 10$ .

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Step 5.6.1.2.1

Multiply $- 4$ by $- 9$ .

$x = \frac{- 13 \pm \sqrt{169 + 36 \cdot - 10}}{2 \cdot - 9}$

Step 5.6.1.2.2

Multiply $36$ by $- 10$ .

$x = \frac{- 13 \pm \sqrt{169 - 360}}{2 \cdot - 9}$

Step 5.6.1.3

Subtract $360$ from $169$ .

$x = \frac{- 13 \pm \sqrt{- 191}}{2 \cdot - 9}$

Step 5.6.1.4

Rewrite $- 191$ as $- 1 (191)$ .

$x = \frac{- 13 \pm \sqrt{- 1 \cdot 191}}{2 \cdot - 9}$

Step 5.6.1.5

Rewrite $\sqrt{- 1 (191)}$ as $\sqrt{- 1} \cdot \sqrt{191}$ .

$x = \frac{- 13 \pm \sqrt{- 1} \cdot \sqrt{191}}{2 \cdot - 9}$

Step 5.6.1.6

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

$x = \frac{- 13 \pm i \sqrt{191}}{2 \cdot - 9}$

Step 5.6.2

Multiply $2$ by $- 9$ .

$x = \frac{- 13 \pm i \sqrt{191}}{- 18}$

Step 5.6.3

Simplify $\frac{- 13 \pm i \sqrt{191}}{- 18}$ .

$x = \frac{13 \pm i \sqrt{191}}{18}$

Step 5.6.4

Change the $\pm$ to $-$ .

$x = \frac{13 - i \sqrt{191}}{18}$

Step 5.7

The final answer is the combination of both solutions.

$x = \frac{13 + i \sqrt{191}}{18}, \frac{13 - i \sqrt{191}}{18}$