Solve 2x*(x+1)+(x-2)*(2x-1/2)=(2x-1/2)^2-7/6x

$$2 x \cdot ( x + 1 ) + ( x - 2 ) \cdot ( 2 x - \frac { 1 } { 2 } ) = ( 2 x - \frac { 1 } { 2 } ) ^ { 2 } - \frac { 7 } { 6 } x$$

Solve for x

$$x = -\frac{9}{8} = -1\frac{1}{8} = -1.125$$

Steps for Solving Linear Equation

Use the distributive property to multiply 2x by x+1.

$$2x^{2}+2x+\left(x-2\right)\left(2x-\frac{1}{2}\right)=\left(2x-\frac{1}{2}\right)^{2}-\frac{7}{6}x$$

Use the distributive property to multiply x-2 by 2x-\frac{1}{2} and combine like terms.

$$2x^{2}+2x+2x^{2}-\frac{9}{2}x+1=\left(2x-\frac{1}{2}\right)^{2}-\frac{7}{6}x$$

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

$$4x^{2}+2x-\frac{9}{2}x+1=\left(2x-\frac{1}{2}\right)^{2}-\frac{7}{6}x$$

Combine 2x and -\frac{9}{2}x to get -\frac{5}{2}x.

$$4x^{2}-\frac{5}{2}x+1=\left(2x-\frac{1}{2}\right)^{2}-\frac{7}{6}x$$

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

$$4x^{2}-\frac{5}{2}x+1=4x^{2}-2x+\frac{1}{4}-\frac{7}{6}x$$

Combine -2x and -\frac{7}{6}x to get -\frac{19}{6}x.

$$4x^{2}-\frac{5}{2}x+1=4x^{2}-\frac{19}{6}x+\frac{1}{4}$$

Subtract 4x^{2} from both sides.

$$4x^{2}-\frac{5}{2}x+1-4x^{2}=-\frac{19}{6}x+\frac{1}{4}$$

Combine 4x^{2} and -4x^{2} to get 0.

$$-\frac{5}{2}x+1=-\frac{19}{6}x+\frac{1}{4}$$

Add \frac{19}{6}x to both sides.

$$-\frac{5}{2}x+1+\frac{19}{6}x=\frac{1}{4}$$

Combine -\frac{5}{2}x and \frac{19}{6}x to get \frac{2}{3}x.

$$\frac{2}{3}x+1=\frac{1}{4}$$

Subtract 1 from both sides.

$$\frac{2}{3}x=\frac{1}{4}-1$$

Subtract 1 from \frac{1}{4} to get -\frac{3}{4}.

$$\frac{2}{3}x=-\frac{3}{4}$$

Multiply both sides by \frac{3}{2}, the reciprocal of \frac{2}{3}.

$$x=-\frac{3}{4}\times \frac{3}{2}$$

Multiply -\frac{3}{4} and \frac{3}{2} to get -\frac{9}{8}.

$$x=-\frac{9}{8}$$