Solve left(m+11right)^2=64

$${ \left(m+11 \right) }^{ 2 } = 64$$

Solve for m

$$m=-3$$

$$m=-19$$

Steps Using Factoring

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(m+11\right)^{2}.

$$m^{2}+22m+121=64$$

Subtract 64 from both sides.

$$m^{2}+22m+121-64=0$$

Subtract 64 from 121 to get 57.

$$m^{2}+22m+57=0$$

To solve the equation, factor m^{2}+22m+57 using formula m^{2}+\left(a+b\right)m+ab=\left(m+a\right)\left(m+b\right). To find a and b, set up a system to be solved.

$$a+b=22 ab=57$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 57.

$$1,57 3,19$$

Calculate the sum for each pair.

$$1+57=58 3+19=22$$

The solution is the pair that gives sum 22.

$$a=3 b=19$$

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

$$\left(m+3\right)\left(m+19\right)$$

To find equation solutions, solve m+3=0 and m+19=0.

$$m=-3 m=-19$$

Steps Using Factoring By Grouping

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(m+11\right)^{2}.

$$m^{2}+22m+121=64$$

Subtract 64 from both sides.

$$m^{2}+22m+121-64=0$$

Subtract 64 from 121 to get 57.

$$m^{2}+22m+57=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as m^{2}+am+bm+57. To find a and b, set up a system to be solved.

$$a+b=22 ab=1\times 57=57$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 57.

$$1,57 3,19$$

Calculate the sum for each pair.

$$1+57=58 3+19=22$$

The solution is the pair that gives sum 22.

$$a=3 b=19$$

Rewrite m^{2}+22m+57 as \left(m^{2}+3m\right)+\left(19m+57\right).

$$\left(m^{2}+3m\right)+\left(19m+57\right)$$

Factor out m in the first and 19 in the second group.

$$m\left(m+3\right)+19\left(m+3\right)$$

Factor out common term m+3 by using distributive property.

$$\left(m+3\right)\left(m+19\right)$$

To find equation solutions, solve m+3=0 and m+19=0.

$$m=-3 m=-19$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(m+11\right)^{2}.

$$m^{2}+22m+121=64$$

Subtract 64 from both sides.

$$m^{2}+22m+121-64=0$$

Subtract 64 from 121 to get 57.

$$m^{2}+22m+57=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 22 for b, and 57 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$m=\frac{-22±\sqrt{22^{2}-4\times 57}}{2}$$

Square 22.

$$m=\frac{-22±\sqrt{484-4\times 57}}{2}$$

Multiply -4 times 57.

$$m=\frac{-22±\sqrt{484-228}}{2}$$

Add 484 to -228.

$$m=\frac{-22±\sqrt{256}}{2}$$

Take the square root of 256.

$$m=\frac{-22±16}{2}$$

Now solve the equation m=\frac{-22±16}{2} when ± is plus. Add -22 to 16.

$$m=-\frac{6}{2}$$

Divide -6 by 2.

$$m=-3$$

Now solve the equation m=\frac{-22±16}{2} when ± is minus. Subtract 16 from -22.

$$m=-\frac{38}{2}$$

Divide -38 by 2.

$$m=-19$$

The equation is now solved.

$$m=-3 m=-19$$

Steps for Completing the Square

Take the square root of both sides of the equation.

$$\sqrt{\left(m+11\right)^{2}}=\sqrt{64}$$

Simplify.

$$m+11=8 m+11=-8$$

Subtract 11 from both sides of the equation.

$$m=-3 m=-19$$