Solve left(x-2right)^2=256

$${ \left(x-2 \right) }^{ 2 } = 256$$

Solve for x

$$x=18$$

$$x=-14$$

Steps Using Factoring

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

$$x^{2}-4x+4=256$$

Subtract 256 from both sides.

$$x^{2}-4x+4-256=0$$

Subtract 256 from 4 to get -252.

$$x^{2}-4x-252=0$$

To solve the equation, factor x^{2}-4x-252 using formula x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). To find a and b, set up a system to be solved.

$$a+b=-4 ab=-252$$

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. List all such integer pairs that give product -252.

$$1,-252 2,-126 3,-84 4,-63 6,-42 7,-36 9,-28 12,-21 14,-18$$

Calculate the sum for each pair.

$$1-252=-251 2-126=-124 3-84=-81 4-63=-59 6-42=-36 7-36=-29 9-28=-19 12-21=-9 14-18=-4$$

The solution is the pair that gives sum -4.

$$a=-18 b=14$$

Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.

$$\left(x-18\right)\left(x+14\right)$$

To find equation solutions, solve x-18=0 and x+14=0.

$$x=18 x=-14$$

Steps Using Factoring By Grouping

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

$$x^{2}-4x+4=256$$

Subtract 256 from both sides.

$$x^{2}-4x+4-256=0$$

Subtract 256 from 4 to get -252.

$$x^{2}-4x-252=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-252. To find a and b, set up a system to be solved.

$$a+b=-4 ab=1\left(-252\right)=-252$$

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. List all such integer pairs that give product -252.

$$1,-252 2,-126 3,-84 4,-63 6,-42 7,-36 9,-28 12,-21 14,-18$$

Calculate the sum for each pair.

$$1-252=-251 2-126=-124 3-84=-81 4-63=-59 6-42=-36 7-36=-29 9-28=-19 12-21=-9 14-18=-4$$

The solution is the pair that gives sum -4.

$$a=-18 b=14$$

Rewrite x^{2}-4x-252 as \left(x^{2}-18x\right)+\left(14x-252\right).

$$\left(x^{2}-18x\right)+\left(14x-252\right)$$

Factor out x in the first and 14 in the second group.

$$x\left(x-18\right)+14\left(x-18\right)$$

Factor out common term x-18 by using distributive property.

$$\left(x-18\right)\left(x+14\right)$$

To find equation solutions, solve x-18=0 and x+14=0.

$$x=18 x=-14$$

Steps Using the Quadratic Formula

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

$$x^{2}-4x+4=256$$

Subtract 256 from both sides.

$$x^{2}-4x+4-256=0$$

Subtract 256 from 4 to get -252.

$$x^{2}-4x-252=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -4 for b, and -252 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\left(-252\right)}}{2}$$

Square -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\left(-252\right)}}{2}$$

Multiply -4 times -252.

$$x=\frac{-\left(-4\right)±\sqrt{16+1008}}{2}$$

Add 16 to 1008.

$$x=\frac{-\left(-4\right)±\sqrt{1024}}{2}$$

Take the square root of 1024.

$$x=\frac{-\left(-4\right)±32}{2}$$

The opposite of -4 is 4.

$$x=\frac{4±32}{2}$$

Now solve the equation x=\frac{4±32}{2} when ± is plus. Add 4 to 32.

$$x=\frac{36}{2}$$

Divide 36 by 2.

$$x=18$$

Now solve the equation x=\frac{4±32}{2} when ± is minus. Subtract 32 from 4.

$$x=-\frac{28}{2}$$

Divide -28 by 2.

$$x=-14$$

The equation is now solved.

$$x=18 x=-14$$

Steps for Completing the Square

Take the square root of both sides of the equation.

$$\sqrt{\left(x-2\right)^{2}}=\sqrt{256}$$

Simplify.

$$x-2=16 x-2=-16$$

Add 2 to both sides of the equation.

$$x=18 x=-14$$