Solve (x+7)^2=121

$$( x + 7 ) ^ { 2 } = 121$$

Solve for x

$$x=4$$

$$x=-18$$

Steps Using Factoring

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+7\right)^{2}.

$$x^{2}+14x+49=121$$

Subtract 121 from both sides.

$$x^{2}+14x+49-121=0$$

Subtract 121 from 49 to get -72.

$$x^{2}+14x-72=0$$

To solve the equation, factor x^{2}+14x-72 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.

$$a+b=14 ab=-72$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -72.

$$-1,72 -2,36 -3,24 -4,18 -6,12 -8,9$$

Calculate the sum for each pair.

$$-1+72=71 -2+36=34 -3+24=21 -4+18=14 -6+12=6 -8+9=1$$

The solution is the pair that gives sum 14.

$$a=-4 b=18$$

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

$$\left(x-4\right)\left(x+18\right)$$

To find equation solutions, solve x-4=0 and x+18=0.

$$x=4 x=-18$$

Steps Using Factoring By Grouping

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+7\right)^{2}.

$$x^{2}+14x+49=121$$

Subtract 121 from both sides.

$$x^{2}+14x+49-121=0$$

Subtract 121 from 49 to get -72.

$$x^{2}+14x-72=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-72. To find a and b, set up a system to be solved.

$$a+b=14 ab=1\left(-72\right)=-72$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -72.

$$-1,72 -2,36 -3,24 -4,18 -6,12 -8,9$$

Calculate the sum for each pair.

$$-1+72=71 -2+36=34 -3+24=21 -4+18=14 -6+12=6 -8+9=1$$

The solution is the pair that gives sum 14.

$$a=-4 b=18$$

Rewrite x^{2}+14x-72 as \left(x^{2}-4x\right)+\left(18x-72\right).

$$\left(x^{2}-4x\right)+\left(18x-72\right)$$

Factor out x in the first and 18 in the second group.

$$x\left(x-4\right)+18\left(x-4\right)$$

Factor out common term x-4 by using distributive property.

$$\left(x-4\right)\left(x+18\right)$$

To find equation solutions, solve x-4=0 and x+18=0.

$$x=4 x=-18$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+7\right)^{2}.

$$x^{2}+14x+49=121$$

Subtract 121 from both sides.

$$x^{2}+14x+49-121=0$$

Subtract 121 from 49 to get -72.

$$x^{2}+14x-72=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 14 for b, and -72 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-14±\sqrt{14^{2}-4\left(-72\right)}}{2}$$

Square 14.

$$x=\frac{-14±\sqrt{196-4\left(-72\right)}}{2}$$

Multiply -4 times -72.

$$x=\frac{-14±\sqrt{196+288}}{2}$$

Add 196 to 288.

$$x=\frac{-14±\sqrt{484}}{2}$$

Take the square root of 484.

$$x=\frac{-14±22}{2}$$

Now solve the equation x=\frac{-14±22}{2} when ± is plus. Add -14 to 22.

$$x=\frac{8}{2}$$

Divide 8 by 2.

$$x=4$$

Now solve the equation x=\frac{-14±22}{2} when ± is minus. Subtract 22 from -14.

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

Divide -36 by 2.

$$x=-18$$

The equation is now solved.

$$x=4 x=-18$$

Steps for Completing the Square

Take the square root of both sides of the equation.

$$\sqrt{\left(x+7\right)^{2}}=\sqrt{121}$$

Simplify.

$$x+7=11 x+7=-11$$

Subtract 7 from both sides of the equation.

$$x=4 x=-18$$