Solve x=frac{-4+7sqrt{3}}{-sqrt{3}-2}

$$x = \frac { - 4 + 7 \sqrt { 3 } } { - \sqrt { 3 } - 2 }$$

Solve for x

$$x=29-18\sqrt{3}\approx -2.176914536$$

Steps for Solving Linear Equation

Rationalize the denominator of \frac{-4+7\sqrt{3}}{-\sqrt{3}-2} by multiplying numerator and denominator by -\sqrt{3}+2.

$$x=\frac{\left(-4+7\sqrt{3}\right)\left(-\sqrt{3}+2\right)}{\left(-\sqrt{3}-2\right)\left(-\sqrt{3}+2\right)}$$

Consider \left(-\sqrt{3}-2\right)\left(-\sqrt{3}+2\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$x=\frac{\left(-4+7\sqrt{3}\right)\left(-\sqrt{3}+2\right)}{\left(-\sqrt{3}\right)^{2}-2^{2}}$$

Calculate -\sqrt{3} to the power of 2 and get \left(\sqrt{3}\right)^{2}.

$$x=\frac{\left(-4+7\sqrt{3}\right)\left(-\sqrt{3}+2\right)}{\left(\sqrt{3}\right)^{2}-2^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$x=\frac{\left(-4+7\sqrt{3}\right)\left(-\sqrt{3}+2\right)}{\left(\sqrt{3}\right)^{2}-4}$$

The square of \sqrt{3} is 3.

$$x=\frac{\left(-4+7\sqrt{3}\right)\left(-\sqrt{3}+2\right)}{3-4}$$

Subtract 4 from 3 to get -1.

$$x=\frac{\left(-4+7\sqrt{3}\right)\left(-\sqrt{3}+2\right)}{-1}$$

Anything divided by -1 gives its opposite.

$$x=-\left(-4+7\sqrt{3}\right)\left(-\sqrt{3}+2\right)$$

Apply the distributive property by multiplying each term of -4+7\sqrt{3} by each term of -\sqrt{3}+2.

$$x=-\left(-4\left(-\sqrt{3}\right)-8+7\sqrt{3}\left(-\sqrt{3}\right)+14\sqrt{3}\right)$$

Multiply -4 and -1 to get 4.

$$x=-\left(4\sqrt{3}-8+7\sqrt{3}\left(-\sqrt{3}\right)+14\sqrt{3}\right)$$

Combine 4\sqrt{3} and 14\sqrt{3} to get 18\sqrt{3}.

$$x=-\left(18\sqrt{3}-8+7\sqrt{3}\left(-\sqrt{3}\right)\right)$$

To find the opposite of 18\sqrt{3}-8+7\sqrt{3}\left(-\sqrt{3}\right), find the opposite of each term.

$$x=-18\sqrt{3}-\left(-8\right)-7\sqrt{3}\left(-\sqrt{3}\right)$$

The opposite of -8 is 8.

$$x=-18\sqrt{3}+8-7\sqrt{3}\left(-\sqrt{3}\right)$$

Multiply -7 and -1 to get 7.

$$x=-18\sqrt{3}+8+7\sqrt{3}\sqrt{3}$$

Multiply \sqrt{3} and \sqrt{3} to get 3.

$$x=-18\sqrt{3}+8+7\times 3$$

Multiply 7 and 3 to get 21.

$$x=-18\sqrt{3}+8+21$$

Add 8 and 21 to get 29.

$$x=-18\sqrt{3}+29$$

Assign x

$$x≔29-18\sqrt{3}$$