Solve (x-6)^2=144

$$( x - 6 ) ^ { 2 } = 144$$

Solve for x

$$x=18$$

$$x=-6$$

Steps Using Factoring

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-6\right)^{2}.

$$x^{2}-12x+36=144$$

Subtract 144 from both sides.

$$x^{2}-12x+36-144=0$$

Subtract 144 from 36 to get -108.

$$x^{2}-12x-108=0$$

To solve the equation, factor x^{2}-12x-108 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.

$$a+b=-12 ab=-108$$

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 -108.

$$1,-108 2,-54 3,-36 4,-27 6,-18 9,-12$$

Calculate the sum for each pair.

$$1-108=-107 2-54=-52 3-36=-33 4-27=-23 6-18=-12 9-12=-3$$

The solution is the pair that gives sum -12.

$$a=-18 b=6$$

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

$$\left(x-18\right)\left(x+6\right)$$

To find equation solutions, solve x-18=0 and x+6=0.

$$x=18 x=-6$$

Steps Using Factoring By Grouping

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-6\right)^{2}.

$$x^{2}-12x+36=144$$

Subtract 144 from both sides.

$$x^{2}-12x+36-144=0$$

Subtract 144 from 36 to get -108.

$$x^{2}-12x-108=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-108. To find a and b, set up a system to be solved.

$$a+b=-12 ab=1\left(-108\right)=-108$$

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 -108.

$$1,-108 2,-54 3,-36 4,-27 6,-18 9,-12$$

Calculate the sum for each pair.

$$1-108=-107 2-54=-52 3-36=-33 4-27=-23 6-18=-12 9-12=-3$$

The solution is the pair that gives sum -12.

$$a=-18 b=6$$

Rewrite x^{2}-12x-108 as \left(x^{2}-18x\right)+\left(6x-108\right).

$$\left(x^{2}-18x\right)+\left(6x-108\right)$$

Factor out x in the first and 6 in the second group.

$$x\left(x-18\right)+6\left(x-18\right)$$

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

$$\left(x-18\right)\left(x+6\right)$$

To find equation solutions, solve x-18=0 and x+6=0.

$$x=18 x=-6$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-6\right)^{2}.

$$x^{2}-12x+36=144$$

Subtract 144 from both sides.

$$x^{2}-12x+36-144=0$$

Subtract 144 from 36 to get -108.

$$x^{2}-12x-108=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -12 for b, and -108 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\left(-108\right)}}{2}$$

Square -12.

$$x=\frac{-\left(-12\right)±\sqrt{144-4\left(-108\right)}}{2}$$

Multiply -4 times -108.

$$x=\frac{-\left(-12\right)±\sqrt{144+432}}{2}$$

Add 144 to 432.

$$x=\frac{-\left(-12\right)±\sqrt{576}}{2}$$

Take the square root of 576.

$$x=\frac{-\left(-12\right)±24}{2}$$

The opposite of -12 is 12.

$$x=\frac{12±24}{2}$$

Now solve the equation x=\frac{12±24}{2} when ± is plus. Add 12 to 24.

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

Divide 36 by 2.

$$x=18$$

Now solve the equation x=\frac{12±24}{2} when ± is minus. Subtract 24 from 12.

$$x=-\frac{12}{2}$$

Divide -12 by 2.

$$x=-6$$

The equation is now solved.

$$x=18 x=-6$$

Steps for Completing the Square

Take the square root of both sides of the equation.

$$\sqrt{\left(x-6\right)^{2}}=\sqrt{144}$$

Simplify.

$$x-6=12 x-6=-12$$

Add 6 to both sides of the equation.

$$x=18 x=-6$$