Solve for x square root of 12( square root of 3-x)=15

$\sqrt{12} (\sqrt{3} - x) = 15$

Step 1

Divide each term in $\sqrt{12} (\sqrt{3} - x) = 15$ by $\sqrt{12}$ and simplify.

Divide each term in $\sqrt{12} (\sqrt{3} - x) = 15$ by $\sqrt{12}$ .

$\frac{\sqrt{12} (\sqrt{3} - x)}{\sqrt{12}} = \frac{15}{\sqrt{12}}$

Simplify the left side.

Cancel the common factor of $\sqrt{12}$ .

Cancel the common factor.

$\frac{\sqrt{12} (\sqrt{3} - x)}{\sqrt{12}} = \frac{15}{\sqrt{12}}$

Divide $\sqrt{3} - x$ by $1$ .

$\sqrt{3} - x = \frac{15}{\sqrt{12}}$

Simplify the right side.

Simplify the denominator.

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

Factor $4$ out of $12$ .

$\sqrt{3} - x = \frac{15}{\sqrt{4 (3)}}$

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

$\sqrt{3} - x = \frac{15}{\sqrt{2^{2} \cdot 3}}$

Pull terms out from under the radical.

$\sqrt{3} - x = \frac{15}{2 \sqrt{3}}$

Multiply $\frac{15}{2 \sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\sqrt{3} - x = \frac{15}{2 \sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Combine and simplify the denominator.

Multiply $\frac{15}{2 \sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\sqrt{3} - x = \frac{15 \sqrt{3}}{2 \sqrt{3} \sqrt{3}}$

Move $\sqrt{3}$ .

$\sqrt{3} - x = \frac{15 \sqrt{3}}{2 (\sqrt{3} \sqrt{3})}$

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

$\sqrt{3} - x = \frac{15 \sqrt{3}}{2 ({\sqrt{3}}^{1} \sqrt{3})}$

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

$\sqrt{3} - x = \frac{15 \sqrt{3}}{2 ({\sqrt{3}}^{1} {\sqrt{3}}^{1})}$

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

$\sqrt{3} - x = \frac{15 \sqrt{3}}{2 {\sqrt{3}}^{1 + 1}}$

Add $1$ and $1$ .

$\sqrt{3} - x = \frac{15 \sqrt{3}}{2 {\sqrt{3}}^{2}}$

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

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

$\sqrt{3} - x = \frac{15 \sqrt{3}}{2 {(3^{\frac{1}{2}})}^{2}}$

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

$\sqrt{3} - x = \frac{15 \sqrt{3}}{2 \cdot 3^{\frac{1}{2} \cdot 2}}$

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

$\sqrt{3} - x = \frac{15 \sqrt{3}}{2 \cdot 3^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\sqrt{3} - x = \frac{15 \sqrt{3}}{2 \cdot 3^{\frac{2}{2}}}$

Rewrite the expression.

$\sqrt{3} - x = \frac{15 \sqrt{3}}{2 \cdot 3^{1}}$

Evaluate the exponent.

$\sqrt{3} - x = \frac{15 \sqrt{3}}{2 \cdot 3}$

Cancel the common factor of $15$ and $3$ .

Factor $3$ out of $15 \sqrt{3}$ .

$\sqrt{3} - x = \frac{3 (5 \sqrt{3})}{2 \cdot 3}$

Cancel the common factors.

Factor $3$ out of $2 \cdot 3$ .

$\sqrt{3} - x = \frac{3 (5 \sqrt{3})}{3 \cdot 2}$

Cancel the common factor.

$\sqrt{3} - x = \frac{3 (5 \sqrt{3})}{3 \cdot 2}$

Rewrite the expression.

$\sqrt{3} - x = \frac{5 \sqrt{3}}{2}$

Step 2

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

Subtract $\sqrt{3}$ from both sides of the equation.

$- x = \frac{5 \sqrt{3}}{2} - \sqrt{3}$

To write $- \sqrt{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- x = \frac{5 \sqrt{3}}{2} - \sqrt{3} \cdot \frac{2}{2}$

Combine $- \sqrt{3}$ and $\frac{2}{2}$ .

$- x = \frac{5 \sqrt{3}}{2} + \frac{- \sqrt{3} \cdot 2}{2}$

Combine the numerators over the common denominator.

$- x = \frac{5 \sqrt{3} - \sqrt{3} \cdot 2}{2}$

Simplify the numerator.

Multiply $2$ by $- 1$ .

$- x = \frac{5 \sqrt{3} - 2 \sqrt{3}}{2}$

Subtract $2 \sqrt{3}$ from $5 \sqrt{3}$ .

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

Step 3

Divide each term in $- x = \frac{3 \sqrt{3}}{2}$ by $- 1$ and simplify.

Divide each term in $- x = \frac{3 \sqrt{3}}{2}$ by $- 1$ .

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

Simplify the left side.

Dividing two negative values results in a positive value.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Move the negative one from the denominator of $\frac{\frac{3 \sqrt{3}}{2}}{- 1}$ .

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

Rewrite $- 1 \cdot \frac{3 \sqrt{3}}{2}$ as $- \frac{3 \sqrt{3}}{2}$ .

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

Step 4

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = - 2.59807621 \dots$

Solve for x square root of 12( square root of 3-x)=15

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