Solve x^4-1/2(x^2-1)^2+1-1/2(x^2+1)

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

Evaluate

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

Solution Steps

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

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

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

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

Use the distributive property to multiply -\frac{1}{2} by x^{4}-2x^{2}+1.

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

Combine x^{4} and -\frac{1}{2}x^{4} to get \frac{1}{2}x^{4}.

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

Add -\frac{1}{2} and 1 to get \frac{1}{2}.

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

Use the distributive property to multiply -\frac{1}{2} by x^{2}+1.

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

Combine x^{2} and -\frac{1}{2}x^{2} to get \frac{1}{2}x^{2}.

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

Subtract \frac{1}{2} from \frac{1}{2} to get 0.

$$\frac{1}{2}x^{4}+\frac{1}{2}x^{2}$$

Expand

$$\frac{x^{4}+x^{2}}{2}$$

Solution Steps

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

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

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

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

Use the distributive property to multiply -\frac{1}{2} by x^{4}-2x^{2}+1.

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

Combine x^{4} and -\frac{1}{2}x^{4} to get \frac{1}{2}x^{4}.

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

Add -\frac{1}{2} and 1 to get \frac{1}{2}.

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

Use the distributive property to multiply -\frac{1}{2} by x^{2}+1.

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

Combine x^{2} and -\frac{1}{2}x^{2} to get \frac{1}{2}x^{2}.

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

Subtract \frac{1}{2} from \frac{1}{2} to get 0.

$$\frac{1}{2}x^{4}+\frac{1}{2}x^{2}$$