Solve (sqrt{3}-5sqrt{2})(sqrt{3}+2)=a+bsqrt{6}

$$( \sqrt { 3 } - 5 \sqrt { 2 } ) ( \sqrt { 3 } + 2 ) = a + b \sqrt { 6 }$$

Solve for b

$$b=-\frac{\sqrt{6}\left(a+5\sqrt{6}+10\sqrt{2}-2\sqrt{3}-3\right)}{6}$$

Steps for Solving Linear Equation

Use the distributive property to multiply \sqrt{3}-5\sqrt{2} by \sqrt{3}+2.

$$\left(\sqrt{3}\right)^{2}+2\sqrt{3}-5\sqrt{2}\sqrt{3}-10\sqrt{2}=a+b\sqrt{6}$$

The square of \sqrt{3} is 3.

$$3+2\sqrt{3}-5\sqrt{2}\sqrt{3}-10\sqrt{2}=a+b\sqrt{6}$$

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

$$3+2\sqrt{3}-5\sqrt{6}-10\sqrt{2}=a+b\sqrt{6}$$

Swap sides so that all variable terms are on the left hand side.

$$a+b\sqrt{6}=3+2\sqrt{3}-5\sqrt{6}-10\sqrt{2}$$

Subtract a from both sides.

$$b\sqrt{6}=3+2\sqrt{3}-5\sqrt{6}-10\sqrt{2}-a$$

The equation is in standard form.

$$\sqrt{6}b=-a+2\sqrt{3}+3-5\sqrt{6}-10\sqrt{2}$$

Divide both sides by \sqrt{6}.

$$\frac{\sqrt{6}b}{\sqrt{6}}=\frac{-a+2\sqrt{3}+3-5\sqrt{6}-10\sqrt{2}}{\sqrt{6}}$$

Dividing by \sqrt{6} undoes the multiplication by \sqrt{6}.

$$b=\frac{-a+2\sqrt{3}+3-5\sqrt{6}-10\sqrt{2}}{\sqrt{6}}$$

Divide 3+2\sqrt{3}-10\sqrt{2}-5\sqrt{6}-a by \sqrt{6}.

$$b=\frac{\sqrt{6}\left(-a+2\sqrt{3}+3-5\sqrt{6}-10\sqrt{2}\right)}{6}$$

Solve for a

$$a=-\sqrt{6}\left(b+5\right)+2\sqrt{3}+3-10\sqrt{2}$$