Algebra Examples

$\sqrt{3} x + \sqrt{12} = \frac{x + 3}{\sqrt{3}}$

Step 1

Multiply both sides by $\sqrt{3}$ .

$(\sqrt{3} x + \sqrt{12}) \sqrt{3} = \frac{x + 3}{\sqrt{3}} \sqrt{3}$

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Simplify $(\sqrt{3} x + \sqrt{12}) \sqrt{3}$ .

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Step 2.1.1.1

Simplify each term.

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Step 2.1.1.1.1

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 2.1.1.1.1.1

Factor $4$ out of $12$ .

$(\sqrt{3} x + \sqrt{4 (3)}) \sqrt{3} = \frac{x + 3}{\sqrt{3}} \sqrt{3}$

Step 2.1.1.1.1.2

Rewrite $4$ as $2^{2}$ .

$(\sqrt{3} x + \sqrt{2^{2} \cdot 3}) \sqrt{3} = \frac{x + 3}{\sqrt{3}} \sqrt{3}$

Step 2.1.1.1.2

Pull terms out from under the radical.

$(\sqrt{3} x + 2 \sqrt{3}) \sqrt{3} = \frac{x + 3}{\sqrt{3}} \sqrt{3}$

Step 2.1.1.2

Apply the distributive property.

$\sqrt{3} x \sqrt{3} + 2 \sqrt{3} \sqrt{3} = \frac{x + 3}{\sqrt{3}} \sqrt{3}$

Step 2.1.1.3

Multiply $\sqrt{3} x \sqrt{3}$ .

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Step 2.1.1.3.1

Raise $\sqrt{3}$ to the power of $1$ .

$x ({\sqrt{3}}^{1} \sqrt{3}) + 2 \sqrt{3} \sqrt{3} = \frac{x + 3}{\sqrt{3}} \sqrt{3}$

Step 2.1.1.3.2

Raise $\sqrt{3}$ to the power of $1$ .

$x ({\sqrt{3}}^{1} {\sqrt{3}}^{1}) + 2 \sqrt{3} \sqrt{3} = \frac{x + 3}{\sqrt{3}} \sqrt{3}$

Step 2.1.1.3.3

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

$x {\sqrt{3}}^{1 + 1} + 2 \sqrt{3} \sqrt{3} = \frac{x + 3}{\sqrt{3}} \sqrt{3}$

Step 2.1.1.3.4

Add $1$ and $1$ .

$x {\sqrt{3}}^{2} + 2 \sqrt{3} \sqrt{3} = \frac{x + 3}{\sqrt{3}} \sqrt{3}$

Step 2.1.1.4

Multiply $2 \sqrt{3} \sqrt{3}$ .

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Step 2.1.1.4.1

Raise $\sqrt{3}$ to the power of $1$ .

$x {\sqrt{3}}^{2} + 2 ({\sqrt{3}}^{1} \sqrt{3}) = \frac{x + 3}{\sqrt{3}} \sqrt{3}$

Step 2.1.1.4.2

Raise $\sqrt{3}$ to the power of $1$ .

$x {\sqrt{3}}^{2} + 2 ({\sqrt{3}}^{1} {\sqrt{3}}^{1}) = \frac{x + 3}{\sqrt{3}} \sqrt{3}$

Step 2.1.1.4.3

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

$x {\sqrt{3}}^{2} + 2 {\sqrt{3}}^{1 + 1} = \frac{x + 3}{\sqrt{3}} \sqrt{3}$

Step 2.1.1.4.4

Add $1$ and $1$ .

$x {\sqrt{3}}^{2} + 2 {\sqrt{3}}^{2} = \frac{x + 3}{\sqrt{3}} \sqrt{3}$

Step 2.1.1.5

Simplify each term.

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Step 2.1.1.5.1

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 2.1.1.5.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$x {(3^{\frac{1}{2}})}^{2} + 2 {\sqrt{3}}^{2} = \frac{x + 3}{\sqrt{3}} \sqrt{3}$

Step 2.1.1.5.1.2

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

$x \cdot 3^{\frac{1}{2} \cdot 2} + 2 {\sqrt{3}}^{2} = \frac{x + 3}{\sqrt{3}} \sqrt{3}$

Step 2.1.1.5.1.3

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

$x \cdot 3^{\frac{2}{2}} + 2 {\sqrt{3}}^{2} = \frac{x + 3}{\sqrt{3}} \sqrt{3}$

Step 2.1.1.5.1.4

Cancel the common factor of $2$ .

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Step 2.1.1.5.1.4.1

Cancel the common factor.

$x \cdot 3^{\frac{2}{2}} + 2 {\sqrt{3}}^{2} = \frac{x + 3}{\sqrt{3}} \sqrt{3}$

Step 2.1.1.5.1.4.2

Rewrite the expression.

$x \cdot 3^{1} + 2 {\sqrt{3}}^{2} = \frac{x + 3}{\sqrt{3}} \sqrt{3}$

Step 2.1.1.5.1.5

Evaluate the exponent.

$x \cdot 3 + 2 {\sqrt{3}}^{2} = \frac{x + 3}{\sqrt{3}} \sqrt{3}$

Step 2.1.1.5.2

Move $3$ to the left of $x$ .

$3 \cdot x + 2 {\sqrt{3}}^{2} = \frac{x + 3}{\sqrt{3}} \sqrt{3}$

Step 2.1.1.5.3

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 2.1.1.5.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$3 x + 2 {(3^{\frac{1}{2}})}^{2} = \frac{x + 3}{\sqrt{3}} \sqrt{3}$

Step 2.1.1.5.3.2

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

$3 x + 2 \cdot 3^{\frac{1}{2} \cdot 2} = \frac{x + 3}{\sqrt{3}} \sqrt{3}$

Step 2.1.1.5.3.3

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

$3 x + 2 \cdot 3^{\frac{2}{2}} = \frac{x + 3}{\sqrt{3}} \sqrt{3}$

Step 2.1.1.5.3.4

Cancel the common factor of $2$ .

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Step 2.1.1.5.3.4.1

Cancel the common factor.

$3 x + 2 \cdot 3^{\frac{2}{2}} = \frac{x + 3}{\sqrt{3}} \sqrt{3}$

Step 2.1.1.5.3.4.2

Rewrite the expression.

$3 x + 2 \cdot 3^{1} = \frac{x + 3}{\sqrt{3}} \sqrt{3}$

Step 2.1.1.5.3.5

Evaluate the exponent.

$3 x + 2 \cdot 3 = \frac{x + 3}{\sqrt{3}} \sqrt{3}$

Step 2.1.1.5.4

Multiply $2$ by $3$ .

$3 x + 6 = \frac{x + 3}{\sqrt{3}} \sqrt{3}$

Step 2.2

Simplify the right side.

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Step 2.2.1

Cancel the common factor of $\sqrt{3}$ .

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Step 2.2.1.1

Cancel the common factor.

$3 x + 6 = \frac{x + 3}{\sqrt{3}} \sqrt{3}$

Step 2.2.1.2

Rewrite the expression.

$3 x + 6 = x + 3$

Step 3

Solve for $x$ .

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Step 3.1

Move all terms containing $x$ to the left side of the equation.

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Step 3.1.1

Subtract $x$ from both sides of the equation.

$3 x + 6 - x = 3$

Step 3.1.2

Subtract $x$ from $3 x$ .

$2 x + 6 = 3$

Step 3.2

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

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Step 3.2.1

Subtract $6$ from both sides of the equation.

$2 x = 3 - 6$

Step 3.2.2

Subtract $6$ from $3$ .

$2 x = - 3$

Step 3.3

Divide each term in $2 x = - 3$ by $2$ and simplify.

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Step 3.3.1

Divide each term in $2 x = - 3$ by $2$ .

$\frac{2 x}{2} = \frac{- 3}{2}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $2$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 3}{2}$

Step 3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 3}{2}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Move the negative in front of the fraction.

$x = - \frac{3}{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{3}{2}$

Decimal Form:

$x = - 1.5$

Mixed Number Form:

$x = - 1 \frac{1}{2}$