Solve (x-3)^2=121

$$( x - 3 ) ^ { 2 } = 121$$

Solve for x

$$x=14$$

$$x=-8$$

Steps Using Factoring

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-3\right)^{2}.

$$x^{2}-6x+9=121$$

Subtract 121 from both sides.

$$x^{2}-6x+9-121=0$$

Subtract 121 from 9 to get -112.

$$x^{2}-6x-112=0$$

To solve the equation, factor x^{2}-6x-112 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.

$$a+b=-6 ab=-112$$

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 -112.

$$1,-112 2,-56 4,-28 7,-16 8,-14$$

Calculate the sum for each pair.

$$1-112=-111 2-56=-54 4-28=-24 7-16=-9 8-14=-6$$

The solution is the pair that gives sum -6.

$$a=-14 b=8$$

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

$$\left(x-14\right)\left(x+8\right)$$

To find equation solutions, solve x-14=0 and x+8=0.

$$x=14 x=-8$$

Steps Using Factoring By Grouping

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-3\right)^{2}.

$$x^{2}-6x+9=121$$

Subtract 121 from both sides.

$$x^{2}-6x+9-121=0$$

Subtract 121 from 9 to get -112.

$$x^{2}-6x-112=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-112. To find a and b, set up a system to be solved.

$$a+b=-6 ab=1\left(-112\right)=-112$$

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 -112.

$$1,-112 2,-56 4,-28 7,-16 8,-14$$

Calculate the sum for each pair.

$$1-112=-111 2-56=-54 4-28=-24 7-16=-9 8-14=-6$$

The solution is the pair that gives sum -6.

$$a=-14 b=8$$

Rewrite x^{2}-6x-112 as \left(x^{2}-14x\right)+\left(8x-112\right).

$$\left(x^{2}-14x\right)+\left(8x-112\right)$$

Factor out x in the first and 8 in the second group.

$$x\left(x-14\right)+8\left(x-14\right)$$

Factor out common term x-14 by using distributive property.

$$\left(x-14\right)\left(x+8\right)$$

To find equation solutions, solve x-14=0 and x+8=0.

$$x=14 x=-8$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-3\right)^{2}.

$$x^{2}-6x+9=121$$

Subtract 121 from both sides.

$$x^{2}-6x+9-121=0$$

Subtract 121 from 9 to get -112.

$$x^{2}-6x-112=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -6 for b, and -112 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\left(-112\right)}}{2}$$

Square -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\left(-112\right)}}{2}$$

Multiply -4 times -112.

$$x=\frac{-\left(-6\right)±\sqrt{36+448}}{2}$$

Add 36 to 448.

$$x=\frac{-\left(-6\right)±\sqrt{484}}{2}$$

Take the square root of 484.

$$x=\frac{-\left(-6\right)±22}{2}$$

The opposite of -6 is 6.

$$x=\frac{6±22}{2}$$

Now solve the equation x=\frac{6±22}{2} when ± is plus. Add 6 to 22.

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

Divide 28 by 2.

$$x=14$$

Now solve the equation x=\frac{6±22}{2} when ± is minus. Subtract 22 from 6.

$$x=-\frac{16}{2}$$

Divide -16 by 2.

$$x=-8$$

The equation is now solved.

$$x=14 x=-8$$

Steps for Completing the Square

Take the square root of both sides of the equation.

$$\sqrt{\left(x-3\right)^{2}}=\sqrt{121}$$

Simplify.

$$x-3=11 x-3=-11$$

Add 3 to both sides of the equation.

$$x=14 x=-8$$