Solve 2(x/2+2)(x-1)+8/5=x^2/2+1/5(4x+1)-3

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

Solve for x

$$x=\frac{\sqrt{101}-11}{5}\approx -0.190024876$$

$$x=\frac{-\sqrt{101}-11}{5}\approx -4.209975124$$

Steps Using the Quadratic Formula

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

$$20\left(\frac{x}{2}+2\right)\left(x-1\right)+16=5x^{2}+2\left(4x+1\right)-30$$

Use the distributive property to multiply 20 by \frac{x}{2}+2.

$$\left(20\times \frac{x}{2}+40\right)\left(x-1\right)+16=5x^{2}+2\left(4x+1\right)-30$$

Cancel out 2, the greatest common factor in 20 and 2.

$$\left(10x+40\right)\left(x-1\right)+16=5x^{2}+2\left(4x+1\right)-30$$

Use the distributive property to multiply 10x+40 by x-1 and combine like terms.

$$10x^{2}+30x-40+16=5x^{2}+2\left(4x+1\right)-30$$

Add -40 and 16 to get -24.

$$10x^{2}+30x-24=5x^{2}+2\left(4x+1\right)-30$$

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

$$10x^{2}+30x-24=5x^{2}+8x+2-30$$

Subtract 30 from 2 to get -28.

$$10x^{2}+30x-24=5x^{2}+8x-28$$

Subtract 5x^{2} from both sides.

$$10x^{2}+30x-24-5x^{2}=8x-28$$

Combine 10x^{2} and -5x^{2} to get 5x^{2}.

$$5x^{2}+30x-24=8x-28$$

Subtract 8x from both sides.

$$5x^{2}+30x-24-8x=-28$$

Combine 30x and -8x to get 22x.

$$5x^{2}+22x-24=-28$$

Add 28 to both sides.

$$5x^{2}+22x-24+28=0$$

Add -24 and 28 to get 4.

$$5x^{2}+22x+4=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, 22 for b, and 4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-22±\sqrt{22^{2}-4\times 5\times 4}}{2\times 5}$$

Square 22.

$$x=\frac{-22±\sqrt{484-4\times 5\times 4}}{2\times 5}$$

Multiply -4 times 5.

$$x=\frac{-22±\sqrt{484-20\times 4}}{2\times 5}$$

Multiply -20 times 4.

$$x=\frac{-22±\sqrt{484-80}}{2\times 5}$$

Add 484 to -80.

$$x=\frac{-22±\sqrt{404}}{2\times 5}$$

Take the square root of 404.

$$x=\frac{-22±2\sqrt{101}}{2\times 5}$$

Multiply 2 times 5.

$$x=\frac{-22±2\sqrt{101}}{10}$$

Now solve the equation x=\frac{-22±2\sqrt{101}}{10} when ± is plus. Add -22 to 2\sqrt{101}.

$$x=\frac{2\sqrt{101}-22}{10}$$

Divide -22+2\sqrt{101} by 10.

$$x=\frac{\sqrt{101}-11}{5}$$

Now solve the equation x=\frac{-22±2\sqrt{101}}{10} when ± is minus. Subtract 2\sqrt{101} from -22.

$$x=\frac{-2\sqrt{101}-22}{10}$$

Divide -22-2\sqrt{101} by 10.

$$x=\frac{-\sqrt{101}-11}{5}$$

The equation is now solved.

$$x=\frac{\sqrt{101}-11}{5} x=\frac{-\sqrt{101}-11}{5}$$

Steps for Completing the Square

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

$$20\left(\frac{x}{2}+2\right)\left(x-1\right)+16=5x^{2}+2\left(4x+1\right)-30$$

Use the distributive property to multiply 20 by \frac{x}{2}+2.

$$\left(20\times \frac{x}{2}+40\right)\left(x-1\right)+16=5x^{2}+2\left(4x+1\right)-30$$

Cancel out 2, the greatest common factor in 20 and 2.

$$\left(10x+40\right)\left(x-1\right)+16=5x^{2}+2\left(4x+1\right)-30$$

Use the distributive property to multiply 10x+40 by x-1 and combine like terms.

$$10x^{2}+30x-40+16=5x^{2}+2\left(4x+1\right)-30$$

Add -40 and 16 to get -24.

$$10x^{2}+30x-24=5x^{2}+2\left(4x+1\right)-30$$

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

$$10x^{2}+30x-24=5x^{2}+8x+2-30$$

Subtract 30 from 2 to get -28.

$$10x^{2}+30x-24=5x^{2}+8x-28$$

Subtract 5x^{2} from both sides.

$$10x^{2}+30x-24-5x^{2}=8x-28$$

Combine 10x^{2} and -5x^{2} to get 5x^{2}.

$$5x^{2}+30x-24=8x-28$$

Subtract 8x from both sides.

$$5x^{2}+30x-24-8x=-28$$

Combine 30x and -8x to get 22x.

$$5x^{2}+22x-24=-28$$

Add 24 to both sides.

$$5x^{2}+22x=-28+24$$

Add -28 and 24 to get -4.

$$5x^{2}+22x=-4$$

Divide both sides by 5.

$$\frac{5x^{2}+22x}{5}=-\frac{4}{5}$$

Dividing by 5 undoes the multiplication by 5.

$$x^{2}+\frac{22}{5}x=-\frac{4}{5}$$

Divide \frac{22}{5}, the coefficient of the x term, by 2 to get \frac{11}{5}. Then add the square of \frac{11}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{22}{5}x+\left(\frac{11}{5}\right)^{2}=-\frac{4}{5}+\left(\frac{11}{5}\right)^{2}$$

Square \frac{11}{5} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{22}{5}x+\frac{121}{25}=-\frac{4}{5}+\frac{121}{25}$$

Add -\frac{4}{5} to \frac{121}{25} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{22}{5}x+\frac{121}{25}=\frac{101}{25}$$

Factor x^{2}+\frac{22}{5}x+\frac{121}{25}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{11}{5}\right)^{2}=\frac{101}{25}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{11}{5}\right)^{2}}=\sqrt{\frac{101}{25}}$$

Simplify.

$$x+\frac{11}{5}=\frac{\sqrt{101}}{5} x+\frac{11}{5}=-\frac{\sqrt{101}}{5}$$

Subtract \frac{11}{5} from both sides of the equation.

$$x=\frac{\sqrt{101}-11}{5} x=\frac{-\sqrt{101}-11}{5}$$