Solve 2(x-1)^2-8=0

$$2 ( x - 1 ) ^ { 2 } - 8 = 0$$

Solve for x

$$x=3$$

$$x=-1$$

Steps Using Factoring By Grouping

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

$$2\left(x^{2}-2x+1\right)-8=0$$

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

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

Subtract 8 from 2 to get -6.

$$2x^{2}-4x-6=0$$

Divide both sides by 2.

$$x^{2}-2x-3=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-3. To find a and b, set up a system to be solved.

$$a+b=-2 ab=1\left(-3\right)=-3$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. The only such pair is the system solution.

$$a=-3 b=1$$

Rewrite x^{2}-2x-3 as \left(x^{2}-3x\right)+\left(x-3\right).

$$\left(x^{2}-3x\right)+\left(x-3\right)$$

Factor out x in x^{2}-3x.

$$x\left(x-3\right)+x-3$$

Factor out common term x-3 by using distributive property.

$$\left(x-3\right)\left(x+1\right)$$

To find equation solutions, solve x-3=0 and x+1=0.

$$x=3 x=-1$$

Steps Using the Quadratic Formula

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

$$2\left(x^{2}-2x+1\right)-8=0$$

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

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

Subtract 8 from 2 to get -6.

$$2x^{2}-4x-6=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -4 for b, and -6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Square -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\times 2\left(-6\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-4\right)±\sqrt{16-8\left(-6\right)}}{2\times 2}$$

Multiply -8 times -6.

$$x=\frac{-\left(-4\right)±\sqrt{16+48}}{2\times 2}$$

Add 16 to 48.

$$x=\frac{-\left(-4\right)±\sqrt{64}}{2\times 2}$$

Take the square root of 64.

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

The opposite of -4 is 4.

$$x=\frac{4±8}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{4±8}{4}$$

Now solve the equation x=\frac{4±8}{4} when ± is plus. Add 4 to 8.

$$x=\frac{12}{4}$$

Divide 12 by 4.

$$x=3$$

Now solve the equation x=\frac{4±8}{4} when ± is minus. Subtract 8 from 4.

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

Divide -4 by 4.

$$x=-1$$

The equation is now solved.

$$x=3 x=-1$$

Steps for Completing the Square

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

$$2\left(x^{2}-2x+1\right)-8=0$$

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

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

Subtract 8 from 2 to get -6.

$$2x^{2}-4x-6=0$$

Add 6 to both sides. Anything plus zero gives itself.

$$2x^{2}-4x=6$$

Divide both sides by 2.

$$\frac{2x^{2}-4x}{2}=\frac{6}{2}$$

Dividing by 2 undoes the multiplication by 2.

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

Divide -4 by 2.

$$x^{2}-2x=\frac{6}{2}$$

Divide 6 by 2.

$$x^{2}-2x=3$$

Divide -2, the coefficient of the x term, by 2 to get -1. Then add the square of -1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-2x+1=3+1$$

Add 3 to 1.

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

Factor x^{2}-2x+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$x-1=2 x-1=-2$$

Add 1 to both sides of the equation.

$$x=3 x=-1$$