Pre-Algebra Examples

$9 x - 3 \sqrt{x - 2} = 0$

Step 1

Subtract $9 x$ from both sides of the equation.

$- 3 \sqrt{x - 2} = - 9 x$

Step 2

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

${(- 3 \sqrt{x - 2})}^{2} = {(- 9 x)}^{2}$

Step 3

Simplify each side of the equation.

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Step 3.1

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

${(- 3 {(x - 2)}^{\frac{1}{2}})}^{2} = {(- 9 x)}^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

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

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Step 3.2.1.1

Apply the product rule to $- 3 {(x - 2)}^{\frac{1}{2}}$ .

${(- 3)}^{2} {({(x - 2)}^{\frac{1}{2}})}^{2} = {(- 9 x)}^{2}$

Step 3.2.1.2

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

$9 {({(x - 2)}^{\frac{1}{2}})}^{2} = {(- 9 x)}^{2}$

Step 3.2.1.3

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

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Step 3.2.1.3.1

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

$9 {(x - 2)}^{\frac{1}{2} \cdot 2} = {(- 9 x)}^{2}$

Step 3.2.1.3.2

Cancel the common factor of $2$ .

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Step 3.2.1.3.2.1

Cancel the common factor.

$9 {(x - 2)}^{\frac{1}{2} \cdot 2} = {(- 9 x)}^{2}$

Step 3.2.1.3.2.2

Rewrite the expression.

$9 {(x - 2)}^{1} = {(- 9 x)}^{2}$

Step 3.2.1.4

Simplify.

$9 (x - 2) = {(- 9 x)}^{2}$

Step 3.2.1.5

Apply the distributive property.

$9 x + 9 \cdot - 2 = {(- 9 x)}^{2}$

Step 3.2.1.6

Multiply $9$ by $- 2$ .

$9 x - 18 = {(- 9 x)}^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify ${(- 9 x)}^{2}$ .

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Step 3.3.1.1

Apply the product rule to $- 9 x$ .

$9 x - 18 = {(- 9)}^{2} x^{2}$

Step 3.3.1.2

Raise $- 9$ to the power of $2$ .

$9 x - 18 = 81 x^{2}$

Step 4

Solve for $x$ .

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Step 4.1

Subtract $81 x^{2}$ from both sides of the equation.

$9 x - 18 - 81 x^{2} = 0$

Step 4.2

Factor $- 9$ out of $9 x - 18 - 81 x^{2}$ .

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Step 4.2.1

Reorder the expression.

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Step 4.2.1.1

Move $- 18$ .

$9 x - 81 x^{2} - 18 = 0$

Step 4.2.1.2

Reorder $9 x$ and $- 81 x^{2}$ .

$- 81 x^{2} + 9 x - 18 = 0$

Step 4.2.2

Factor $- 9$ out of $- 81 x^{2}$ .

$- 9 (9 x^{2}) + 9 x - 18 = 0$

Step 4.2.3

Factor $- 9$ out of $9 x$ .

$- 9 (9 x^{2}) - 9 (- x) - 18 = 0$

Step 4.2.4

Factor $- 9$ out of $- 18$ .

$- 9 (9 x^{2}) - 9 (- x) - 9 \cdot 2 = 0$

Step 4.2.5

Factor $- 9$ out of $- 9 (9 x^{2}) - 9 (- x)$ .

$- 9 (9 x^{2} - x) - 9 \cdot 2 = 0$

Step 4.2.6

Factor $- 9$ out of $- 9 (9 x^{2} - x) - 9 (2)$ .

$- 9 (9 x^{2} - x + 2) = 0$

Step 4.3

Divide each term in $- 9 (9 x^{2} - x + 2) = 0$ by $- 9$ and simplify.

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Step 4.3.1

Divide each term in $- 9 (9 x^{2} - x + 2) = 0$ by $- 9$ .

$\frac{- 9 (9 x^{2} - x + 2)}{- 9} = \frac{0}{- 9}$

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Cancel the common factor of $- 9$ .

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Step 4.3.2.1.1

Cancel the common factor.

$\frac{- 9 (9 x^{2} - x + 2)}{- 9} = \frac{0}{- 9}$

Step 4.3.2.1.2

Divide $9 x^{2} - x + 2$ by $1$ .

$9 x^{2} - x + 2 = \frac{0}{- 9}$

Step 4.3.3

Simplify the right side.

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Step 4.3.3.1

Divide $0$ by $- 9$ .

$9 x^{2} - x + 2 = 0$

Step 4.4

Use the quadratic formula to find the solutions.

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

Step 4.5

Substitute the values $a = 9$ , $b = - 1$ , and $c = 2$ into the quadratic formula and solve for $x$ .

$\frac{1 \pm \sqrt{{(- 1)}^{2} - 4 \cdot (9 \cdot 2)}}{2 \cdot 9}$

Step 4.6

Simplify.

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Step 4.6.1

Simplify the numerator.

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Step 4.6.1.1

Raise $- 1$ to the power of $2$ .

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

Step 4.6.1.2

Multiply $- 4 \cdot 9 \cdot 2$ .

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Step 4.6.1.2.1

Multiply $- 4$ by $9$ .

$x = \frac{1 \pm \sqrt{1 - 36 \cdot 2}}{2 \cdot 9}$

Step 4.6.1.2.2

Multiply $- 36$ by $2$ .

$x = \frac{1 \pm \sqrt{1 - 72}}{2 \cdot 9}$

Step 4.6.1.3

Subtract $72$ from $1$ .

$x = \frac{1 \pm \sqrt{- 71}}{2 \cdot 9}$

Step 4.6.1.4

Rewrite $- 71$ as $- 1 (71)$ .

$x = \frac{1 \pm \sqrt{- 1 \cdot 71}}{2 \cdot 9}$

Step 4.6.1.5

Rewrite $\sqrt{- 1 (71)}$ as $\sqrt{- 1} \cdot \sqrt{71}$ .

$x = \frac{1 \pm \sqrt{- 1} \cdot \sqrt{71}}{2 \cdot 9}$

Step 4.6.1.6

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

$x = \frac{1 \pm i \sqrt{71}}{2 \cdot 9}$

Step 4.6.2

Multiply $2$ by $9$ .

$x = \frac{1 \pm i \sqrt{71}}{18}$

Step 4.7

The final answer is the combination of both solutions.

$x = \frac{1 + i \sqrt{71}}{18}, \frac{1 - i \sqrt{71}}{18}$