Solve frac{4+sqrt{2}}{2+sqrt{2}}=a-bsqrt{2}

$$\frac { 4 + \sqrt { 2 } } { 2 + \sqrt { 2 } } = a - b \sqrt { 2 }$$

Solve for b

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

Steps for Solving Linear Equation

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

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

Consider \left(2+\sqrt{2}\right)\left(2-\sqrt{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}.

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

Square 2. Square \sqrt{2}.

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

Subtract 2 from 4 to get 2.

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

Use the distributive property to multiply 4+\sqrt{2} by 2-\sqrt{2} and combine like terms.

$$\frac{8-2\sqrt{2}-\left(\sqrt{2}\right)^{2}}{2}=a-b\sqrt{2}$$

The square of \sqrt{2} is 2.

$$\frac{8-2\sqrt{2}-2}{2}=a-b\sqrt{2}$$

Subtract 2 from 8 to get 6.

$$\frac{6-2\sqrt{2}}{2}=a-b\sqrt{2}$$

Divide each term of 6-2\sqrt{2} by 2 to get 3-\sqrt{2}.

$$3-\sqrt{2}=a-b\sqrt{2}$$

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

$$a-b\sqrt{2}=3-\sqrt{2}$$

Subtract a from both sides.

$$-b\sqrt{2}=3-\sqrt{2}-a$$

The equation is in standard form.

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

Divide both sides by -\sqrt{2}.

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

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

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

Divide 3-\sqrt{2}-a by -\sqrt{2}.

$$b=\frac{\sqrt{2}a}{2}-\frac{3\sqrt{2}}{2}+1$$

Solve for a

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