Solve alpha^2+beta^2=(alpha+beta)^2-2

$$\alpha ^ { 2 } + \beta ^ { 2 } = ( \alpha + \beta ) ^ { 2 } - 2$$

Solve for α

$$\alpha =\frac{1}{\beta }$$

$$\beta \neq 0$$

Steps for Solving Linear Equation

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

$$\alpha ^{2}+\beta ^{2}=\alpha ^{2}+2\alpha \beta +\beta ^{2}-2$$

Subtract \alpha ^{2} from both sides.

$$\alpha ^{2}+\beta ^{2}-\alpha ^{2}=2\alpha \beta +\beta ^{2}-2$$

Combine \alpha ^{2} and -\alpha ^{2} to get 0.

$$\beta ^{2}=2\alpha \beta +\beta ^{2}-2$$

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

$$2\alpha \beta +\beta ^{2}-2=\beta ^{2}$$

Subtract \beta ^{2} from both sides.

$$2\alpha \beta -2=\beta ^{2}-\beta ^{2}$$

Combine \beta ^{2} and -\beta ^{2} to get 0.

$$2\alpha \beta -2=0$$

Add 2 to both sides. Anything plus zero gives itself.

$$2\alpha \beta =2$$

The equation is in standard form.

$$2\beta \alpha =2$$

Divide both sides by 2\beta .

$$\frac{2\beta \alpha }{2\beta }=\frac{2}{2\beta }$$

Dividing by 2\beta undoes the multiplication by 2\beta .

$$\alpha =\frac{2}{2\beta }$$

Divide 2 by 2\beta .

$$\alpha =\frac{1}{\beta }$$

Solve for β

$$\beta =\frac{1}{\alpha }$$

$$\alpha \neq 0$$

Steps for Solving Linear Equation

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

$$\alpha ^{2}+\beta ^{2}=\alpha ^{2}+2\alpha \beta +\beta ^{2}-2$$

Subtract 2\alpha \beta from both sides.

$$\alpha ^{2}+\beta ^{2}-2\alpha \beta =\alpha ^{2}+\beta ^{2}-2$$

Subtract \beta ^{2} from both sides.

$$\alpha ^{2}+\beta ^{2}-2\alpha \beta -\beta ^{2}=\alpha ^{2}-2$$

Combine \beta ^{2} and -\beta ^{2} to get 0.

$$\alpha ^{2}-2\alpha \beta =\alpha ^{2}-2$$

Subtract \alpha ^{2} from both sides.

$$-2\alpha \beta =\alpha ^{2}-2-\alpha ^{2}$$

Combine \alpha ^{2} and -\alpha ^{2} to get 0.

$$-2\alpha \beta =-2$$

The equation is in standard form.

$$\left(-2\alpha \right)\beta =-2$$

Divide both sides by -2\alpha .

$$\frac{\left(-2\alpha \right)\beta }{-2\alpha }=-\frac{2}{-2\alpha }$$

Dividing by -2\alpha undoes the multiplication by -2\alpha .

$$\beta =-\frac{2}{-2\alpha }$$

Divide -2 by -2\alpha .

$$\beta =\frac{1}{\alpha }$$