Algebra Examples

$\frac{x}{\sqrt{3}} = d - x$

Step 1

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

$\frac{x}{\sqrt{3}} \sqrt{3} = (d - x) \sqrt{3}$

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

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

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Step 2.1.1.1

Cancel the common factor.

$\frac{x}{\sqrt{3}} \sqrt{3} = (d - x) \sqrt{3}$

Step 2.1.1.2

Rewrite the expression.

$x = (d - x) \sqrt{3}$

Step 2.2

Simplify the right side.

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Step 2.2.1

Apply the distributive property.

$x = d \sqrt{3} - x \sqrt{3}$

Step 3

Solve for $x$ .

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Step 3.1

Add $x \sqrt{3}$ to both sides of the equation.

$x + x \sqrt{3} = d \sqrt{3}$

Step 3.2

Factor $x$ out of $x + x \sqrt{3}$ .

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Step 3.2.1

Factor $x$ out of $x^{1}$ .

$x \cdot 1 + x \sqrt{3} = d \sqrt{3}$

Step 3.2.2

Factor $x$ out of $x \sqrt{3}$ .

$x \cdot 1 + x (\sqrt{3}) = d \sqrt{3}$

Step 3.2.3

Factor $x$ out of $x \cdot 1 + x (\sqrt{3})$ .

$x (1 + \sqrt{3}) = d \sqrt{3}$

Step 3.3

Divide each term in $x (1 + \sqrt{3}) = d \sqrt{3}$ by $1 + \sqrt{3}$ and simplify.

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Step 3.3.1

Divide each term in $x (1 + \sqrt{3}) = d \sqrt{3}$ by $1 + \sqrt{3}$ .

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

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $1 + \sqrt{3}$ .

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Step 3.3.2.1.1

Cancel the common factor.

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

Step 3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{d \sqrt{3}}{1 + \sqrt{3}}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Multiply $\frac{d \sqrt{3}}{1 + \sqrt{3}}$ by $\frac{1 - \sqrt{3}}{1 - \sqrt{3}}$ .

$x = \frac{d \sqrt{3}}{1 + \sqrt{3}} \cdot \frac{1 - \sqrt{3}}{1 - \sqrt{3}}$

Step 3.3.3.2

Multiply $\frac{d \sqrt{3}}{1 + \sqrt{3}}$ by $\frac{1 - \sqrt{3}}{1 - \sqrt{3}}$ .

$x = \frac{d \sqrt{3} (1 - \sqrt{3})}{(1 + \sqrt{3}) (1 - \sqrt{3})}$

Step 3.3.3.3

Expand the denominator using the FOIL method.

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

Step 3.3.3.4

Simplify.

$x = \frac{d \sqrt{3} (1 - \sqrt{3})}{- 2}$

Step 3.3.3.5

Group $1 - \sqrt{3}$ and $\sqrt{3}$ together.

$x = \frac{d ((1 - \sqrt{3}) \sqrt{3})}{- 2}$

Step 3.3.3.6

Apply the distributive property.

$x = \frac{d (1 \sqrt{3} - \sqrt{3} \sqrt{3})}{- 2}$

Step 3.3.3.7

Multiply $\sqrt{3}$ by $1$ .

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

Step 3.3.3.8

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

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Step 3.3.3.8.1

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

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

Step 3.3.3.8.2

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

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

Step 3.3.3.8.3

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

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

Step 3.3.3.8.4

Add $1$ and $1$ .

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

Step 3.3.3.9

Simplify each term.

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Step 3.3.3.9.1

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

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Step 3.3.3.9.1.1

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

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

Step 3.3.3.9.1.2

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

$x = \frac{d (\sqrt{3} - 3^{\frac{1}{2} \cdot 2})}{- 2}$

Step 3.3.3.9.1.3

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

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

Step 3.3.3.9.1.4

Cancel the common factor of $2$ .

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Step 3.3.3.9.1.4.1

Cancel the common factor.

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

Step 3.3.3.9.1.4.2

Rewrite the expression.

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

Step 3.3.3.9.1.5

Evaluate the exponent.

$x = \frac{d (\sqrt{3} - 1 \cdot 3)}{- 2}$

Step 3.3.3.9.2

Multiply $- 1$ by $3$ .

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

Step 3.3.3.10

Move the negative in front of the fraction.

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