Solve {l}{a=frac{sqrt{2}+1}{sqrt{2}-1}}{b=frac{sqrt{2}-1}{sqrt{2}+1}}

$$\left. \begin{array} { l } { a = \frac { \sqrt { 2 } + 1 } { \sqrt { 2 } - 1 } } \\ { b = \frac { \sqrt { 2 } - 1 } { \sqrt { 2 } + 1 } } \end{array} \right.$$

Solve for a, b

$$a=2\sqrt{2}+3\approx 5.828427125$$

$$b=3-2\sqrt{2}\approx 0.171572875$$

Solution Steps

Consider the first equation. Rationalize the denominator of \frac{\sqrt{2}+1}{\sqrt{2}-1} by multiplying numerator and denominator by \sqrt{2}+1.

$$a=\frac{\left(\sqrt{2}+1\right)\left(\sqrt{2}+1\right)}{\left(\sqrt{2}-1\right)\left(\sqrt{2}+1\right)}$$

Consider \left(\sqrt{2}-1\right)\left(\sqrt{2}+1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$a=\frac{\left(\sqrt{2}+1\right)\left(\sqrt{2}+1\right)}{\left(\sqrt{2}\right)^{2}-1^{2}}$$

Square \sqrt{2}. Square 1.

$$a=\frac{\left(\sqrt{2}+1\right)\left(\sqrt{2}+1\right)}{2-1}$$

Subtract 1 from 2 to get 1.

$$a=\frac{\left(\sqrt{2}+1\right)\left(\sqrt{2}+1\right)}{1}$$

Anything divided by one gives itself.

$$a=\left(\sqrt{2}+1\right)\left(\sqrt{2}+1\right)$$

Multiply \sqrt{2}+1 and \sqrt{2}+1 to get \left(\sqrt{2}+1\right)^{2}.

$$a=\left(\sqrt{2}+1\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\sqrt{2}+1\right)^{2}.

$$a=\left(\sqrt{2}\right)^{2}+2\sqrt{2}+1$$

The square of \sqrt{2} is 2.

$$a=2+2\sqrt{2}+1$$

Add 2 and 1 to get 3.

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

Consider the second equation. Rationalize the denominator of \frac{\sqrt{2}-1}{\sqrt{2}+1} by multiplying numerator and denominator by \sqrt{2}-1.

$$b=\frac{\left(\sqrt{2}-1\right)\left(\sqrt{2}-1\right)}{\left(\sqrt{2}+1\right)\left(\sqrt{2}-1\right)}$$

Consider \left(\sqrt{2}+1\right)\left(\sqrt{2}-1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$b=\frac{\left(\sqrt{2}-1\right)\left(\sqrt{2}-1\right)}{\left(\sqrt{2}\right)^{2}-1^{2}}$$

Square \sqrt{2}. Square 1.

$$b=\frac{\left(\sqrt{2}-1\right)\left(\sqrt{2}-1\right)}{2-1}$$

Subtract 1 from 2 to get 1.

$$b=\frac{\left(\sqrt{2}-1\right)\left(\sqrt{2}-1\right)}{1}$$

Anything divided by one gives itself.

$$b=\left(\sqrt{2}-1\right)\left(\sqrt{2}-1\right)$$

Multiply \sqrt{2}-1 and \sqrt{2}-1 to get \left(\sqrt{2}-1\right)^{2}.

$$b=\left(\sqrt{2}-1\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\sqrt{2}-1\right)^{2}.

$$b=\left(\sqrt{2}\right)^{2}-2\sqrt{2}+1$$

The square of \sqrt{2} is 2.

$$b=2-2\sqrt{2}+1$$

Add 2 and 1 to get 3.

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

The system is now solved.

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