Solve frac{2sqrt{2}-3sqrt{7}}{2sqrt{2}+3sqrt{7}}=a+bsqrt{2}

$$\frac { 2 \sqrt { 2 } - 3 \sqrt { 7 } } { 2 \sqrt { 2 } + 3 \sqrt { 7 } } = a + b \sqrt { 2 }$$

Solve for b

$$b=-\frac{\sqrt{2}\left(55a+71-12\sqrt{14}\right)}{110}$$

Steps for Solving Linear Equation

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

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

Consider \left(2\sqrt{2}+3\sqrt{7}\right)\left(2\sqrt{2}-3\sqrt{7}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Multiply 2\sqrt{2}-3\sqrt{7} and 2\sqrt{2}-3\sqrt{7} to get \left(2\sqrt{2}-3\sqrt{7}\right)^{2}.

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(2\sqrt{2}-3\sqrt{7}\right)^{2}.

$$\frac{4\left(\sqrt{2}\right)^{2}-12\sqrt{2}\sqrt{7}+9\left(\sqrt{7}\right)^{2}}{\left(2\sqrt{2}\right)^{2}-\left(3\sqrt{7}\right)^{2}}=a+b\sqrt{2}$$

The square of \sqrt{2} is 2.

$$\frac{4\times 2-12\sqrt{2}\sqrt{7}+9\left(\sqrt{7}\right)^{2}}{\left(2\sqrt{2}\right)^{2}-\left(3\sqrt{7}\right)^{2}}=a+b\sqrt{2}$$

Multiply 4 and 2 to get 8.

$$\frac{8-12\sqrt{2}\sqrt{7}+9\left(\sqrt{7}\right)^{2}}{\left(2\sqrt{2}\right)^{2}-\left(3\sqrt{7}\right)^{2}}=a+b\sqrt{2}$$

To multiply \sqrt{2} and \sqrt{7}, multiply the numbers under the square root.

$$\frac{8-12\sqrt{14}+9\left(\sqrt{7}\right)^{2}}{\left(2\sqrt{2}\right)^{2}-\left(3\sqrt{7}\right)^{2}}=a+b\sqrt{2}$$

The square of \sqrt{7} is 7.

$$\frac{8-12\sqrt{14}+9\times 7}{\left(2\sqrt{2}\right)^{2}-\left(3\sqrt{7}\right)^{2}}=a+b\sqrt{2}$$

Multiply 9 and 7 to get 63.

$$\frac{8-12\sqrt{14}+63}{\left(2\sqrt{2}\right)^{2}-\left(3\sqrt{7}\right)^{2}}=a+b\sqrt{2}$$

Add 8 and 63 to get 71.

$$\frac{71-12\sqrt{14}}{\left(2\sqrt{2}\right)^{2}-\left(3\sqrt{7}\right)^{2}}=a+b\sqrt{2}$$

Expand \left(2\sqrt{2}\right)^{2}.

$$\frac{71-12\sqrt{14}}{2^{2}\left(\sqrt{2}\right)^{2}-\left(3\sqrt{7}\right)^{2}}=a+b\sqrt{2}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{71-12\sqrt{14}}{4\left(\sqrt{2}\right)^{2}-\left(3\sqrt{7}\right)^{2}}=a+b\sqrt{2}$$

The square of \sqrt{2} is 2.

$$\frac{71-12\sqrt{14}}{4\times 2-\left(3\sqrt{7}\right)^{2}}=a+b\sqrt{2}$$

Multiply 4 and 2 to get 8.

$$\frac{71-12\sqrt{14}}{8-\left(3\sqrt{7}\right)^{2}}=a+b\sqrt{2}$$

Expand \left(3\sqrt{7}\right)^{2}.

$$\frac{71-12\sqrt{14}}{8-3^{2}\left(\sqrt{7}\right)^{2}}=a+b\sqrt{2}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{71-12\sqrt{14}}{8-9\left(\sqrt{7}\right)^{2}}=a+b\sqrt{2}$$

The square of \sqrt{7} is 7.

$$\frac{71-12\sqrt{14}}{8-9\times 7}=a+b\sqrt{2}$$

Multiply 9 and 7 to get 63.

$$\frac{71-12\sqrt{14}}{8-63}=a+b\sqrt{2}$$

Subtract 63 from 8 to get -55.

$$\frac{71-12\sqrt{14}}{-55}=a+b\sqrt{2}$$

Multiply both numerator and denominator by -1.

$$\frac{-71+12\sqrt{14}}{55}=a+b\sqrt{2}$$

Divide each term of -71+12\sqrt{14} by 55 to get -\frac{71}{55}+\frac{12}{55}\sqrt{14}.

$$-\frac{71}{55}+\frac{12}{55}\sqrt{14}=a+b\sqrt{2}$$

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

$$a+b\sqrt{2}=-\frac{71}{55}+\frac{12}{55}\sqrt{14}$$

Subtract a from both sides.

$$b\sqrt{2}=-\frac{71}{55}+\frac{12}{55}\sqrt{14}-a$$

The equation is in standard form.

$$\sqrt{2}b=-a+\frac{12\sqrt{14}}{55}-\frac{71}{55}$$

Divide both sides by \sqrt{2}.

$$\frac{\sqrt{2}b}{\sqrt{2}}=\frac{-a+\frac{12\sqrt{14}}{55}-\frac{71}{55}}{\sqrt{2}}$$

Dividing by \sqrt{2} undoes the multiplication by \sqrt{2}.

$$b=\frac{-a+\frac{12\sqrt{14}}{55}-\frac{71}{55}}{\sqrt{2}}$$

Divide -\frac{71}{55}+\frac{12\sqrt{14}}{55}-a by \sqrt{2}.

$$b=\frac{\sqrt{2}\left(-55a+12\sqrt{14}-71\right)}{110}$$

Solve for a

$$a=-\frac{\left(2\sqrt{2}-3\sqrt{7}\right)\left(-3\sqrt{14}b-4b+2\sqrt{2}-3\sqrt{7}\right)}{55}$$