Solve x=frac{4sqrt{3}-6sqrt{2}}{sqrt{2}+sqrt{3}}*frac{(sqrt{2}-sqrt{3})}{(sqrt{2}-sqrt{3})}

$$x = \frac { 4 \sqrt { 3 } - 6 \sqrt { 2 } } { \sqrt { 2 } + \sqrt { 3 } } \cdot \frac { ( \sqrt { 2 } - \sqrt { 3 } ) } { ( \sqrt { 2 } - \sqrt { 3 } ) }$$

Solve for x

$$x=24-10\sqrt{6}\approx -0.494897428$$

Steps for Solving Linear Equation

Rationalize the denominator of \frac{4\sqrt{3}-6\sqrt{2}}{\sqrt{2}+\sqrt{3}} by multiplying numerator and denominator by \sqrt{2}-\sqrt{3}.

$$x=\frac{\left(4\sqrt{3}-6\sqrt{2}\right)\left(\sqrt{2}-\sqrt{3}\right)}{\left(\sqrt{2}+\sqrt{3}\right)\left(\sqrt{2}-\sqrt{3}\right)}\times \frac{\sqrt{2}-\sqrt{3}}{\sqrt{2}-\sqrt{3}}$$

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

$$x=\frac{\left(4\sqrt{3}-6\sqrt{2}\right)\left(\sqrt{2}-\sqrt{3}\right)}{\left(\sqrt{2}\right)^{2}-\left(\sqrt{3}\right)^{2}}\times \frac{\sqrt{2}-\sqrt{3}}{\sqrt{2}-\sqrt{3}}$$

Square \sqrt{2}. Square \sqrt{3}.

$$x=\frac{\left(4\sqrt{3}-6\sqrt{2}\right)\left(\sqrt{2}-\sqrt{3}\right)}{2-3}\times \frac{\sqrt{2}-\sqrt{3}}{\sqrt{2}-\sqrt{3}}$$

Subtract 3 from 2 to get -1.

$$x=\frac{\left(4\sqrt{3}-6\sqrt{2}\right)\left(\sqrt{2}-\sqrt{3}\right)}{-1}\times \frac{\sqrt{2}-\sqrt{3}}{\sqrt{2}-\sqrt{3}}$$

Anything divided by -1 gives its opposite.

$$x=\left(-\left(4\sqrt{3}-6\sqrt{2}\right)\left(\sqrt{2}-\sqrt{3}\right)\right)\times \frac{\sqrt{2}-\sqrt{3}}{\sqrt{2}-\sqrt{3}}$$

Cancel out \sqrt{2}-\sqrt{3} in both numerator and denominator.

$$x=\left(-\left(4\sqrt{3}-6\sqrt{2}\right)\left(\sqrt{2}-\sqrt{3}\right)\right)\times 1$$

Apply the distributive property by multiplying each term of 4\sqrt{3}-6\sqrt{2} by each term of \sqrt{2}-\sqrt{3}.

$$x=\left(-\left(4\sqrt{3}\sqrt{2}-4\left(\sqrt{3}\right)^{2}-6\left(\sqrt{2}\right)^{2}+6\sqrt{3}\sqrt{2}\right)\right)\times 1$$

To multiply \sqrt{3} and \sqrt{2}, multiply the numbers under the square root.

$$x=\left(-\left(4\sqrt{6}-4\left(\sqrt{3}\right)^{2}-6\left(\sqrt{2}\right)^{2}+6\sqrt{3}\sqrt{2}\right)\right)\times 1$$

The square of \sqrt{3} is 3.

$$x=\left(-\left(4\sqrt{6}-4\times 3-6\left(\sqrt{2}\right)^{2}+6\sqrt{3}\sqrt{2}\right)\right)\times 1$$

Multiply -4 and 3 to get -12.

$$x=\left(-\left(4\sqrt{6}-12-6\left(\sqrt{2}\right)^{2}+6\sqrt{3}\sqrt{2}\right)\right)\times 1$$

The square of \sqrt{2} is 2.

$$x=\left(-\left(4\sqrt{6}-12-6\times 2+6\sqrt{3}\sqrt{2}\right)\right)\times 1$$

Multiply -6 and 2 to get -12.

$$x=\left(-\left(4\sqrt{6}-12-12+6\sqrt{3}\sqrt{2}\right)\right)\times 1$$

Subtract 12 from -12 to get -24.

$$x=\left(-\left(4\sqrt{6}-24+6\sqrt{3}\sqrt{2}\right)\right)\times 1$$

To multiply \sqrt{3} and \sqrt{2}, multiply the numbers under the square root.

$$x=\left(-\left(4\sqrt{6}-24+6\sqrt{6}\right)\right)\times 1$$

Combine 4\sqrt{6} and 6\sqrt{6} to get 10\sqrt{6}.

$$x=\left(-\left(10\sqrt{6}-24\right)\right)\times 1$$

To find the opposite of 10\sqrt{6}-24, find the opposite of each term.

$$x=\left(-10\sqrt{6}-\left(-24\right)\right)\times 1$$

The opposite of -24 is 24.

$$x=\left(-10\sqrt{6}+24\right)\times 1$$

Use the distributive property to multiply -10\sqrt{6}+24 by 1.

$$x=-10\sqrt{6}+24$$

Assign x

$$x≔24-10\sqrt{6}$$