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

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

Solve for x

$$x=\sqrt{2}+2\approx 3.414213562$$

$$x=2-\sqrt{2}\approx 0.585786438$$

Steps Using the Quadratic Formula

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

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

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

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

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

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

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

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

Combine \frac{1}{2}x^{2} and \frac{1}{2}x^{2} to get x^{2}.

$$x^{2}-2-2x+2=2x-1-1$$

Add -2 and 2 to get 0.

$$x^{2}-2x=2x-1-1$$

Subtract 1 from -1 to get -2.

$$x^{2}-2x=2x-2$$

Subtract 2x from both sides.

$$x^{2}-2x-2x=-2$$

Combine -2x and -2x to get -4x.

$$x^{2}-4x=-2$$

Add 2 to both sides.

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

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

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

Square -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\times 2}}{2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-4\right)±\sqrt{16-8}}{2}$$

Add 16 to -8.

$$x=\frac{-\left(-4\right)±\sqrt{8}}{2}$$

Take the square root of 8.

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

The opposite of -4 is 4.

$$x=\frac{4±2\sqrt{2}}{2}$$

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

$$x=\frac{2\sqrt{2}+4}{2}$$

Divide 4+2\sqrt{2} by 2.

$$x=\sqrt{2}+2$$

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

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

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

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

The equation is now solved.

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

Steps for Completing the Square

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

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

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

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

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

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

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

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

Combine \frac{1}{2}x^{2} and \frac{1}{2}x^{2} to get x^{2}.

$$x^{2}-2-2x+2=2x-1-1$$

Add -2 and 2 to get 0.

$$x^{2}-2x=2x-1-1$$

Subtract 1 from -1 to get -2.

$$x^{2}-2x=2x-2$$

Subtract 2x from both sides.

$$x^{2}-2x-2x=-2$$

Combine -2x and -2x to get -4x.

$$x^{2}-4x=-2$$

Divide -4, the coefficient of the x term, by 2 to get -2. Then add the square of -2 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square -2.

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

Add -2 to 4.

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

Factor x^{2}-4x+4. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-2\right)^{2}=2$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-2\right)^{2}}=\sqrt{2}$$

Simplify.

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

Add 2 to both sides of the equation.

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