Solve alpha^{4}+beta^{4}=(alpha^2+beta^2)^2-2(alphabeta)^2

$$\alpha ^ { 4 } + \beta ^ { 4 } = ( \alpha ^ { 2 } + \beta ^ { 2 } ) ^ { 2 } - 2 ( \alpha \beta ) ^ { 2 }$$

Solve for α (complex solution)

$$\alpha \in \mathrm{C}$$

Solution Steps

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

$$\alpha ^{4}+\beta ^{4}=\left(\alpha ^{2}\right)^{2}+2\alpha ^{2}\beta ^{2}+\left(\beta ^{2}\right)^{2}-2\left(\alpha \beta \right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$\alpha ^{4}+\beta ^{4}=\alpha ^{4}+2\alpha ^{2}\beta ^{2}+\left(\beta ^{2}\right)^{2}-2\left(\alpha \beta \right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$\alpha ^{4}+\beta ^{4}=\alpha ^{4}+2\alpha ^{2}\beta ^{2}+\beta ^{4}-2\left(\alpha \beta \right)^{2}$$

Expand \left(\alpha \beta \right)^{2}.

$$\alpha ^{4}+\beta ^{4}=\alpha ^{4}+2\alpha ^{2}\beta ^{2}+\beta ^{4}-2\alpha ^{2}\beta ^{2}$$

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

$$\alpha ^{4}+\beta ^{4}=\alpha ^{4}+\beta ^{4}$$

Subtract \alpha ^{4} from both sides.

$$\alpha ^{4}+\beta ^{4}-\alpha ^{4}=\beta ^{4}$$

Combine \alpha ^{4} and -\alpha ^{4} to get 0.

$$\beta ^{4}=\beta ^{4}$$

Reorder the terms.

$$\text{true}$$

This is true for any \alpha .

$$\alpha \in \mathrm{C}$$

Solve for β (complex solution)

$$\beta \in \mathrm{C}$$

Solution Steps

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

$$\alpha ^{4}+\beta ^{4}=\left(\alpha ^{2}\right)^{2}+2\alpha ^{2}\beta ^{2}+\left(\beta ^{2}\right)^{2}-2\left(\alpha \beta \right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$\alpha ^{4}+\beta ^{4}=\alpha ^{4}+2\alpha ^{2}\beta ^{2}+\left(\beta ^{2}\right)^{2}-2\left(\alpha \beta \right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$\alpha ^{4}+\beta ^{4}=\alpha ^{4}+2\alpha ^{2}\beta ^{2}+\beta ^{4}-2\left(\alpha \beta \right)^{2}$$

Expand \left(\alpha \beta \right)^{2}.

$$\alpha ^{4}+\beta ^{4}=\alpha ^{4}+2\alpha ^{2}\beta ^{2}+\beta ^{4}-2\alpha ^{2}\beta ^{2}$$

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

$$\alpha ^{4}+\beta ^{4}=\alpha ^{4}+\beta ^{4}$$

Subtract \beta ^{4} from both sides.

$$\alpha ^{4}+\beta ^{4}-\beta ^{4}=\alpha ^{4}$$

Combine \beta ^{4} and -\beta ^{4} to get 0.

$$\alpha ^{4}=\alpha ^{4}$$

Reorder the terms.

$$\text{true}$$

This is true for any \beta .

$$\beta \in \mathrm{C}$$

Solve for α

$$\alpha \in \mathrm{R}$$

Solution Steps

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

$$\alpha ^{4}+\beta ^{4}=\left(\alpha ^{2}\right)^{2}+2\alpha ^{2}\beta ^{2}+\left(\beta ^{2}\right)^{2}-2\left(\alpha \beta \right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$\alpha ^{4}+\beta ^{4}=\alpha ^{4}+2\alpha ^{2}\beta ^{2}+\left(\beta ^{2}\right)^{2}-2\left(\alpha \beta \right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$\alpha ^{4}+\beta ^{4}=\alpha ^{4}+2\alpha ^{2}\beta ^{2}+\beta ^{4}-2\left(\alpha \beta \right)^{2}$$

Expand \left(\alpha \beta \right)^{2}.

$$\alpha ^{4}+\beta ^{4}=\alpha ^{4}+2\alpha ^{2}\beta ^{2}+\beta ^{4}-2\alpha ^{2}\beta ^{2}$$

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

$$\alpha ^{4}+\beta ^{4}=\alpha ^{4}+\beta ^{4}$$

Subtract \alpha ^{4} from both sides.

$$\alpha ^{4}+\beta ^{4}-\alpha ^{4}=\beta ^{4}$$

Combine \alpha ^{4} and -\alpha ^{4} to get 0.

$$\beta ^{4}=\beta ^{4}$$

Reorder the terms.

$$\text{true}$$

This is true for any \alpha .

$$\alpha \in \mathrm{R}$$

Solve for β

$$\beta \in \mathrm{R}$$

Solution Steps

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

$$\alpha ^{4}+\beta ^{4}=\left(\alpha ^{2}\right)^{2}+2\alpha ^{2}\beta ^{2}+\left(\beta ^{2}\right)^{2}-2\left(\alpha \beta \right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$\alpha ^{4}+\beta ^{4}=\alpha ^{4}+2\alpha ^{2}\beta ^{2}+\left(\beta ^{2}\right)^{2}-2\left(\alpha \beta \right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$\alpha ^{4}+\beta ^{4}=\alpha ^{4}+2\alpha ^{2}\beta ^{2}+\beta ^{4}-2\left(\alpha \beta \right)^{2}$$

Expand \left(\alpha \beta \right)^{2}.

$$\alpha ^{4}+\beta ^{4}=\alpha ^{4}+2\alpha ^{2}\beta ^{2}+\beta ^{4}-2\alpha ^{2}\beta ^{2}$$

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

$$\alpha ^{4}+\beta ^{4}=\alpha ^{4}+\beta ^{4}$$

Subtract \beta ^{4} from both sides.

$$\alpha ^{4}+\beta ^{4}-\beta ^{4}=\alpha ^{4}$$

Combine \beta ^{4} and -\beta ^{4} to get 0.

$$\alpha ^{4}=\alpha ^{4}$$

Reorder the terms.

$$\text{true}$$

This is true for any \beta .

$$\beta \in \mathrm{R}$$