Algebra Examples

$\frac{12}{\sqrt{x + 2}} + 8 = 10$

Step 1

Solve for $\sqrt{x + 2}$ .

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Step 1.1

Move all terms not containing $\sqrt{x + 2}$ to the right side of the equation.

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Step 1.1.1

Subtract $8$ from both sides of the equation.

$\frac{12}{\sqrt{x + 2}} = 10 - 8$

Step 1.1.2

Subtract $8$ from $10$ .

$\frac{12}{\sqrt{x + 2}} = 2$

Step 1.2

Find the LCD of the terms in the equation.

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Step 1.2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$\sqrt{x + 2}, 1$

Step 1.2.2

The LCM of one and any expression is the expression.

$\sqrt{x + 2}$

Step 1.3

Multiply each term in $\frac{12}{\sqrt{x + 2}} = 2$ by $\sqrt{x + 2}$ to eliminate the fractions.

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Step 1.3.1

Multiply each term in $\frac{12}{\sqrt{x + 2}} = 2$ by $\sqrt{x + 2}$ .

$\frac{12}{\sqrt{x + 2}} \sqrt{x + 2} = 2 \sqrt{x + 2}$

Step 1.3.2

Simplify the left side.

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Step 1.3.2.1

Cancel the common factor of $\sqrt{x + 2}$ .

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Step 1.3.2.1.1

Cancel the common factor.

$\frac{12}{\sqrt{x + 2}} \sqrt{x + 2} = 2 \sqrt{x + 2}$

Step 1.3.2.1.2

Rewrite the expression.

$12 = 2 \sqrt{x + 2}$

Step 1.4

Solve the equation.

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Step 1.4.1

Rewrite the equation as $2 \sqrt{x + 2} = 12$ .

$2 \sqrt{x + 2} = 12$

Step 1.4.2

Divide each term in $2 \sqrt{x + 2} = 12$ by $2$ and simplify.

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Step 1.4.2.1

Divide each term in $2 \sqrt{x + 2} = 12$ by $2$ .

$\frac{2 \sqrt{x + 2}}{2} = \frac{12}{2}$

Step 1.4.2.2

Simplify the left side.

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Step 1.4.2.2.1

Cancel the common factor of $2$ .

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Step 1.4.2.2.1.1

Cancel the common factor.

$\frac{2 \sqrt{x + 2}}{2} = \frac{12}{2}$

Step 1.4.2.2.1.2

Divide $\sqrt{x + 2}$ by $1$ .

$\sqrt{x + 2} = \frac{12}{2}$

Step 1.4.2.3

Simplify the right side.

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Step 1.4.2.3.1

Divide $12$ by $2$ .

$\sqrt{x + 2} = 6$

Step 2

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

${\sqrt{x + 2}}^{2} = 6^{2}$

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

${({(x + 2)}^{\frac{1}{2}})}^{2} = 6^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

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

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Step 3.2.1.1

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

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Step 3.2.1.1.1

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

${(x + 2)}^{\frac{1}{2} \cdot 2} = 6^{2}$

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

${(x + 2)}^{\frac{1}{2} \cdot 2} = 6^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(x + 2)}^{1} = 6^{2}$

Step 3.2.1.2

Simplify.

$x + 2 = 6^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Raise $6$ to the power of $2$ .

$x + 2 = 36$

Step 4

Move all terms not containing $x$ to the right side of the equation.

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Step 4.1

Subtract $2$ from both sides of the equation.

$x = 36 - 2$

Step 4.2

Subtract $2$ from $36$ .

$x = 34$