Algebra Examples

$\sqrt{4 x} + \sqrt{6} = 12$

Step 1

Subtract $\sqrt{6}$ from both sides of the equation.

$\sqrt{4 x} = 12 - \sqrt{6}$

Step 2

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

${\sqrt{4 x}}^{2} = {(12 - \sqrt{6})}^{2}$

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 $\sqrt{4 x}$ as ${(4 x)}^{\frac{1}{2}}$ .

${({(4 x)}^{\frac{1}{2}})}^{2} = {(12 - \sqrt{6})}^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${({(4 x)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Multiply the exponents in ${({(4 x)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1.1

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

${(4 x)}^{\frac{1}{2} \cdot 2} = {(12 - \sqrt{6})}^{2}$

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.

${(4 x)}^{\frac{1}{2} \cdot 2} = {(12 - \sqrt{6})}^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(4 x)}^{1} = {(12 - \sqrt{6})}^{2}$

Step 3.2.1.2

Simplify.

$4 x = {(12 - \sqrt{6})}^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify ${(12 - \sqrt{6})}^{2}$ .

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Step 3.3.1.1

Rewrite ${(12 - \sqrt{6})}^{2}$ as $(12 - \sqrt{6}) (12 - \sqrt{6})$ .

$4 x = (12 - \sqrt{6}) (12 - \sqrt{6})$

Step 3.3.1.2

Expand $(12 - \sqrt{6}) (12 - \sqrt{6})$ using the FOIL Method.

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Step 3.3.1.2.1

Apply the distributive property.

$4 x = 12 (12 - \sqrt{6}) - \sqrt{6} (12 - \sqrt{6})$

Step 3.3.1.2.2

Apply the distributive property.

$4 x = 12 \cdot 12 + 12 (- \sqrt{6}) - \sqrt{6} (12 - \sqrt{6})$

Step 3.3.1.2.3

Apply the distributive property.

$4 x = 12 \cdot 12 + 12 (- \sqrt{6}) - \sqrt{6} \cdot 12 - \sqrt{6} (- \sqrt{6})$

Step 3.3.1.3

Simplify and combine like terms.

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Step 3.3.1.3.1

Simplify each term.

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Step 3.3.1.3.1.1

Multiply $12$ by $12$ .

$4 x = 144 + 12 (- \sqrt{6}) - \sqrt{6} \cdot 12 - \sqrt{6} (- \sqrt{6})$

Step 3.3.1.3.1.2

Multiply $- 1$ by $12$ .

$4 x = 144 - 12 \sqrt{6} - \sqrt{6} \cdot 12 - \sqrt{6} (- \sqrt{6})$

Step 3.3.1.3.1.3

Multiply $12$ by $- 1$ .

$4 x = 144 - 12 \sqrt{6} - 12 \sqrt{6} - \sqrt{6} (- \sqrt{6})$

Step 3.3.1.3.1.4

Multiply $- \sqrt{6} (- \sqrt{6})$ .

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Step 3.3.1.3.1.4.1

Multiply $- 1$ by $- 1$ .

$4 x = 144 - 12 \sqrt{6} - 12 \sqrt{6} + 1 \sqrt{6} \sqrt{6}$

Step 3.3.1.3.1.4.2

Multiply $\sqrt{6}$ by $1$ .

$4 x = 144 - 12 \sqrt{6} - 12 \sqrt{6} + \sqrt{6} \sqrt{6}$

Step 3.3.1.3.1.4.3

Raise $\sqrt{6}$ to the power of $1$ .

$4 x = 144 - 12 \sqrt{6} - 12 \sqrt{6} + {\sqrt{6}}^{1} \sqrt{6}$

Step 3.3.1.3.1.4.4

Raise $\sqrt{6}$ to the power of $1$ .

$4 x = 144 - 12 \sqrt{6} - 12 \sqrt{6} + {\sqrt{6}}^{1} {\sqrt{6}}^{1}$

Step 3.3.1.3.1.4.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$4 x = 144 - 12 \sqrt{6} - 12 \sqrt{6} + {\sqrt{6}}^{1 + 1}$

Step 3.3.1.3.1.4.6

Add $1$ and $1$ .

$4 x = 144 - 12 \sqrt{6} - 12 \sqrt{6} + {\sqrt{6}}^{2}$

Step 3.3.1.3.1.5

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 3.3.1.3.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$4 x = 144 - 12 \sqrt{6} - 12 \sqrt{6} + {(6^{\frac{1}{2}})}^{2}$

Step 3.3.1.3.1.5.2

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

$4 x = 144 - 12 \sqrt{6} - 12 \sqrt{6} + 6^{\frac{1}{2} \cdot 2}$

Step 3.3.1.3.1.5.3

Combine $\frac{1}{2}$ and $2$ .

$4 x = 144 - 12 \sqrt{6} - 12 \sqrt{6} + 6^{\frac{2}{2}}$

Step 3.3.1.3.1.5.4

Cancel the common factor of $2$ .

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Step 3.3.1.3.1.5.4.1

Cancel the common factor.

$4 x = 144 - 12 \sqrt{6} - 12 \sqrt{6} + 6^{\frac{2}{2}}$

Step 3.3.1.3.1.5.4.2

Rewrite the expression.

$4 x = 144 - 12 \sqrt{6} - 12 \sqrt{6} + 6^{1}$

Step 3.3.1.3.1.5.5

Evaluate the exponent.

$4 x = 144 - 12 \sqrt{6} - 12 \sqrt{6} + 6$

Step 3.3.1.3.2

Add $144$ and $6$ .

$4 x = 150 - 12 \sqrt{6} - 12 \sqrt{6}$

Step 3.3.1.3.3

Subtract $12 \sqrt{6}$ from $- 12 \sqrt{6}$ .

$4 x = 150 - 24 \sqrt{6}$

Step 4

Divide each term in $4 x = 150 - 24 \sqrt{6}$ by $4$ and simplify.

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Step 4.1

Divide each term in $4 x = 150 - 24 \sqrt{6}$ by $4$ .

$\frac{4 x}{4} = \frac{150}{4} + \frac{- 24 \sqrt{6}}{4}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $4$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{150}{4} + \frac{- 24 \sqrt{6}}{4}$

Step 4.2.1.2

Divide $x$ by $1$ .

$x = \frac{150}{4} + \frac{- 24 \sqrt{6}}{4}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Simplify each term.

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Step 4.3.1.1

Cancel the common factor of $150$ and $4$ .

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Step 4.3.1.1.1

Factor $2$ out of $150$ .

$x = \frac{2 (75)}{4} + \frac{- 24 \sqrt{6}}{4}$

Step 4.3.1.1.2

Cancel the common factors.

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Step 4.3.1.1.2.1

Factor $2$ out of $4$ .

$x = \frac{2 \cdot 75}{2 \cdot 2} + \frac{- 24 \sqrt{6}}{4}$

Step 4.3.1.1.2.2

Cancel the common factor.

$x = \frac{2 \cdot 75}{2 \cdot 2} + \frac{- 24 \sqrt{6}}{4}$

Step 4.3.1.1.2.3

Rewrite the expression.

$x = \frac{75}{2} + \frac{- 24 \sqrt{6}}{4}$

Step 4.3.1.2

Cancel the common factor of $- 24$ and $4$ .

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Step 4.3.1.2.1

Factor $4$ out of $- 24 \sqrt{6}$ .

$x = \frac{75}{2} + \frac{4 (- 6 \sqrt{6})}{4}$

Step 4.3.1.2.2

Cancel the common factors.

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Step 4.3.1.2.2.1

Factor $4$ out of $4$ .

$x = \frac{75}{2} + \frac{4 (- 6 \sqrt{6})}{4 (1)}$

Step 4.3.1.2.2.2

Cancel the common factor.

$x = \frac{75}{2} + \frac{4 (- 6 \sqrt{6})}{4 \cdot 1}$

Step 4.3.1.2.2.3

Rewrite the expression.

$x = \frac{75}{2} + \frac{- 6 \sqrt{6}}{1}$

Step 4.3.1.2.2.4

Divide $- 6 \sqrt{6}$ by $1$ .

$x = \frac{75}{2} - 6 \sqrt{6}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{75}{2} - 6 \sqrt{6}$

Decimal Form:

$x = 22.80306154 \dots$