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

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

Solve for x

$$x=\frac{1}{9}\approx 0.111111111$$

Steps for Solving Linear Equation

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

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\frac{3}{2}-x\right)^{2}.

$$x^{2}-x+\frac{1}{4}+\frac{9}{4}-3x+x^{2}-2x\left(x+\frac{1}{4}\right)=2$$

Add \frac{1}{4} and \frac{9}{4} to get \frac{5}{2}.

$$x^{2}-x+\frac{5}{2}-3x+x^{2}-2x\left(x+\frac{1}{4}\right)=2$$

Combine -x and -3x to get -4x.

$$x^{2}-4x+\frac{5}{2}+x^{2}-2x\left(x+\frac{1}{4}\right)=2$$

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

$$2x^{2}-4x+\frac{5}{2}-2x\left(x+\frac{1}{4}\right)=2$$

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

$$2x^{2}-4x+\frac{5}{2}-2x^{2}-\frac{1}{2}x=2$$

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

$$-4x+\frac{5}{2}-\frac{1}{2}x=2$$

Combine -4x and -\frac{1}{2}x to get -\frac{9}{2}x.

$$-\frac{9}{2}x+\frac{5}{2}=2$$

Subtract \frac{5}{2} from both sides.

$$-\frac{9}{2}x=2-\frac{5}{2}$$

Subtract \frac{5}{2} from 2 to get -\frac{1}{2}.

$$-\frac{9}{2}x=-\frac{1}{2}$$

Multiply both sides by -\frac{2}{9}, the reciprocal of -\frac{9}{2}.

$$x=-\frac{1}{2}\left(-\frac{2}{9}\right)$$

Multiply -\frac{1}{2} and -\frac{2}{9} to get \frac{1}{9}.

$$x=\frac{1}{9}$$