Solve (x^2-1)^2+(1-x^2)^2=

$$( x ^ { 2 } - 1 ) ^ { 2 } + ( 1 - x ^ { 2 } ) ^ { 2 } =$$

Evaluate

$$2\left(x^{2}-1\right)^{2}$$

Solution Steps

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

$$\left(x^{2}\right)^{2}-2x^{2}+1+\left(1-x^{2}\right)^{2}$$

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

$$x^{4}-2x^{2}+1+\left(1-x^{2}\right)^{2}$$

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

$$x^{4}-2x^{2}+1+1-2x^{2}+\left(x^{2}\right)^{2}$$

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

$$x^{4}-2x^{2}+1+1-2x^{2}+x^{4}$$

Add 1 and 1 to get 2.

$$x^{4}-2x^{2}+2-2x^{2}+x^{4}$$

Combine -2x^{2} and -2x^{2} to get -4x^{2}.

$$x^{4}-4x^{2}+2+x^{4}$$

Combine x^{4} and x^{4} to get 2x^{4}.

$$2x^{4}-4x^{2}+2$$

Expand

$$2x^{4}-4x^{2}+2$$

Solution Steps

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

$$\left(x^{2}\right)^{2}-2x^{2}+1+\left(1-x^{2}\right)^{2}$$

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

$$x^{4}-2x^{2}+1+\left(1-x^{2}\right)^{2}$$

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

$$x^{4}-2x^{2}+1+1-2x^{2}+\left(x^{2}\right)^{2}$$

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

$$x^{4}-2x^{2}+1+1-2x^{2}+x^{4}$$

Add 1 and 1 to get 2.

$$x^{4}-2x^{2}+2-2x^{2}+x^{4}$$

Combine -2x^{2} and -2x^{2} to get -4x^{2}.

$$x^{4}-4x^{2}+2+x^{4}$$

Combine x^{4} and x^{4} to get 2x^{4}.

$$2x^{4}-4x^{2}+2$$