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

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

Evaluate

$$-6x^{2}+5x-\frac{3}{8}$$

Solution Steps

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

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

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

$$2x^{2}+x+\frac{1}{8}-2\left(2x-\frac{1}{2}\right)^{2}$$

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

$$2x^{2}+x+\frac{1}{8}-2\left(4x^{2}-2x+\frac{1}{4}\right)$$

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

$$2x^{2}+x+\frac{1}{8}-8x^{2}+4x-\frac{1}{2}$$

Combine 2x^{2} and -8x^{2} to get -6x^{2}.

$$-6x^{2}+x+\frac{1}{8}+4x-\frac{1}{2}$$

Combine x and 4x to get 5x.

$$-6x^{2}+5x+\frac{1}{8}-\frac{1}{2}$$

Subtract \frac{1}{2} from \frac{1}{8} to get -\frac{3}{8}.

$$-6x^{2}+5x-\frac{3}{8}$$

Expand

$$-6x^{2}+5x-\frac{3}{8}$$

Solution Steps

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

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

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

$$2x^{2}+x+\frac{1}{8}-2\left(2x-\frac{1}{2}\right)^{2}$$

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

$$2x^{2}+x+\frac{1}{8}-2\left(4x^{2}-2x+\frac{1}{4}\right)$$

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

$$2x^{2}+x+\frac{1}{8}-8x^{2}+4x-\frac{1}{2}$$

Combine 2x^{2} and -8x^{2} to get -6x^{2}.

$$-6x^{2}+x+\frac{1}{8}+4x-\frac{1}{2}$$

Combine x and 4x to get 5x.

$$-6x^{2}+5x+\frac{1}{8}-\frac{1}{2}$$

Subtract \frac{1}{2} from \frac{1}{8} to get -\frac{3}{8}.

$$-6x^{2}+5x-\frac{3}{8}$$