Solve (x+2)*(2x-2)-x^2div2-(x-2)*(x-6)=2

$$(x+2) \times (2x-2)- { x }^{ 2 } \div 2-(x-2) \times (x-6)=2$$

Solve for x

$$x=2\sqrt{34}-10\approx 1.66190379$$

$$x=-2\sqrt{34}-10\approx -21.66190379$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 2.

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

Use the distributive property to multiply x+2 by 2x-2 and combine like terms.

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

Combine 2x^{2} and -\frac{x^{2}}{2} to get \frac{3}{2}x^{2}.

$$2\left(\frac{3}{2}x^{2}+2x-4\right)-2\left(x-2\right)\left(x-6\right)=4$$

Use the distributive property to multiply 2 by \frac{3}{2}x^{2}+2x-4.

$$3x^{2}+4x-8-2\left(x-2\right)\left(x-6\right)=4$$

Subtract 4 from both sides.

$$3x^{2}+4x-8-2\left(x-2\right)\left(x-6\right)-4=0$$

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

$$3x^{2}+4x-8+\left(-2x+4\right)\left(x-6\right)-4=0$$

Use the distributive property to multiply -2x+4 by x-6 and combine like terms.

$$3x^{2}+4x-8-2x^{2}+16x-24-4=0$$

Combine 3x^{2} and -2x^{2} to get x^{2}.

$$x^{2}+4x-8+16x-24-4=0$$

Combine 4x and 16x to get 20x.

$$x^{2}+20x-8-24-4=0$$

Subtract 24 from -8 to get -32.

$$x^{2}+20x-32-4=0$$

Subtract 4 from -32 to get -36.

$$x^{2}+20x-36=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 20 for b, and -36 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-20±\sqrt{20^{2}-4\left(-36\right)}}{2}$$

Square 20.

$$x=\frac{-20±\sqrt{400-4\left(-36\right)}}{2}$$

Multiply -4 times -36.

$$x=\frac{-20±\sqrt{400+144}}{2}$$

Add 400 to 144.

$$x=\frac{-20±\sqrt{544}}{2}$$

Take the square root of 544.

$$x=\frac{-20±4\sqrt{34}}{2}$$

Now solve the equation x=\frac{-20±4\sqrt{34}}{2} when ± is plus. Add -20 to 4\sqrt{34}.

$$x=\frac{4\sqrt{34}-20}{2}$$

Divide -20+4\sqrt{34} by 2.

$$x=2\sqrt{34}-10$$

Now solve the equation x=\frac{-20±4\sqrt{34}}{2} when ± is minus. Subtract 4\sqrt{34} from -20.

$$x=\frac{-4\sqrt{34}-20}{2}$$

Divide -20-4\sqrt{34} by 2.

$$x=-2\sqrt{34}-10$$

The equation is now solved.

$$x=2\sqrt{34}-10 x=-2\sqrt{34}-10$$

Steps for Completing the Square

Multiply both sides of the equation by 2.

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

Use the distributive property to multiply x+2 by 2x-2 and combine like terms.

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

Combine 2x^{2} and -\frac{x^{2}}{2} to get \frac{3}{2}x^{2}.

$$2\left(\frac{3}{2}x^{2}+2x-4\right)-2\left(x-2\right)\left(x-6\right)=4$$

Use the distributive property to multiply 2 by \frac{3}{2}x^{2}+2x-4.

$$3x^{2}+4x-8-2\left(x-2\right)\left(x-6\right)=4$$

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

$$3x^{2}+4x-8+\left(-2x+4\right)\left(x-6\right)=4$$

Use the distributive property to multiply -2x+4 by x-6 and combine like terms.

$$3x^{2}+4x-8-2x^{2}+16x-24=4$$

Combine 3x^{2} and -2x^{2} to get x^{2}.

$$x^{2}+4x-8+16x-24=4$$

Combine 4x and 16x to get 20x.

$$x^{2}+20x-8-24=4$$

Subtract 24 from -8 to get -32.

$$x^{2}+20x-32=4$$

Add 32 to both sides.

$$x^{2}+20x=4+32$$

Add 4 and 32 to get 36.

$$x^{2}+20x=36$$

Divide 20, the coefficient of the x term, by 2 to get 10. Then add the square of 10 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+20x+10^{2}=36+10^{2}$$

Square 10.

$$x^{2}+20x+100=36+100$$

Add 36 to 100.

$$x^{2}+20x+100=136$$

Factor x^{2}+20x+100. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+10\right)^{2}=136$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+10\right)^{2}}=\sqrt{136}$$

Simplify.

$$x+10=2\sqrt{34} x+10=-2\sqrt{34}$$

Subtract 10 from both sides of the equation.

$$x=2\sqrt{34}-10 x=-2\sqrt{34}-10$$