Solve (1/2-x)^2+3x=1/4+x(x+2)

$$( \frac { 1 } { 2 } - x ) ^ { 2 } + 3 x = \frac { 1 } { 4 } + x ( x + 2 )$$

Solve for x (complex solution)

$$x\in \mathrm{C}$$

Solution Steps

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

$$\frac{1}{4}-x+x^{2}+3x=\frac{1}{4}+x\left(x+2\right)$$

Combine -x and 3x to get 2x.

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

Use the distributive property to multiply x by x+2.

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

Subtract \frac{1}{4} from both sides.

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

Subtract \frac{1}{4} from \frac{1}{4} to get 0.

$$2x+x^{2}=x^{2}+2x$$

Subtract x^{2} from both sides.

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

Combine x^{2} and -x^{2} to get 0.

$$2x=2x$$

Subtract 2x from both sides.

$$2x-2x=0$$

Combine 2x and -2x to get 0.

$$0=0$$

Compare 0 and 0.

$$\text{true}$$

This is true for any x.

$$x\in \mathrm{C}$$

Solve for x

$$x\in \mathrm{R}$$

Solution Steps

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

$$\frac{1}{4}-x+x^{2}+3x=\frac{1}{4}+x\left(x+2\right)$$

Combine -x and 3x to get 2x.

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

Use the distributive property to multiply x by x+2.

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

Subtract \frac{1}{4} from both sides.

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

Subtract \frac{1}{4} from \frac{1}{4} to get 0.

$$2x+x^{2}=x^{2}+2x$$

Subtract x^{2} from both sides.

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

Combine x^{2} and -x^{2} to get 0.

$$2x=2x$$

Subtract 2x from both sides.

$$2x-2x=0$$

Combine 2x and -2x to get 0.

$$0=0$$

Compare 0 and 0.

$$\text{true}$$

This is true for any x.

$$x\in \mathrm{R}$$