Solve (x/2+2)(x-1)+8/5=x^2/2+1/5(4x+1)-frac{3}{10}(x-1)-frac{2}{5}

$$( \frac { x } { 2 } + 2 ) ( x - 1 ) + \frac { 8 } { 5 } = \frac { x ^ { 2 } } { 2 } + \frac { 1 } { 5 } ( 4 x + 1 ) - \frac { 3 } { 10 } ( x - 1 ) - \frac { 2 } { 5 }$$

Solve for x

$$x=\frac{1}{2}=0.5$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 10, the least common multiple of 2,5,10.

$$10\left(\frac{x}{2}+2\right)\left(x-1\right)+16=5x^{2}+2\left(4x+1\right)-3\left(x-1\right)-4$$

Use the distributive property to multiply 10 by \frac{x}{2}+2.

$$\left(10\times \frac{x}{2}+20\right)\left(x-1\right)+16=5x^{2}+2\left(4x+1\right)-3\left(x-1\right)-4$$

Cancel out 2, the greatest common factor in 10 and 2.

$$\left(5x+20\right)\left(x-1\right)+16=5x^{2}+2\left(4x+1\right)-3\left(x-1\right)-4$$

Use the distributive property to multiply 5x+20 by x-1 and combine like terms.

$$5x^{2}+15x-20+16=5x^{2}+2\left(4x+1\right)-3\left(x-1\right)-4$$

Add -20 and 16 to get -4.

$$5x^{2}+15x-4=5x^{2}+2\left(4x+1\right)-3\left(x-1\right)-4$$

Use the distributive property to multiply 2 by 4x+1.

$$5x^{2}+15x-4=5x^{2}+8x+2-3\left(x-1\right)-4$$

Use the distributive property to multiply -3 by x-1.

$$5x^{2}+15x-4=5x^{2}+8x+2-3x+3-4$$

Combine 8x and -3x to get 5x.

$$5x^{2}+15x-4=5x^{2}+5x+2+3-4$$

Add 2 and 3 to get 5.

$$5x^{2}+15x-4=5x^{2}+5x+5-4$$

Subtract 4 from 5 to get 1.

$$5x^{2}+15x-4=5x^{2}+5x+1$$

Subtract 5x^{2} from both sides.

$$5x^{2}+15x-4-5x^{2}=5x+1$$

Combine 5x^{2} and -5x^{2} to get 0.

$$15x-4=5x+1$$

Subtract 5x from both sides.

$$15x-4-5x=1$$

Combine 15x and -5x to get 10x.

$$10x-4=1$$

Add 4 to both sides.

$$10x=1+4$$

Add 1 and 4 to get 5.

$$10x=5$$

Divide both sides by 10.

$$x=\frac{5}{10}$$

Reduce the fraction \frac{5}{10} to lowest terms by extracting and canceling out 5.

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