Solve (b+12)^2=81

$$( b + 12 ) ^ { 2 } = 81$$

Solve for b

$$b=-3$$

$$b=-21$$

Steps Using Factoring

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(b+12\right)^{2}.

$$b^{2}+24b+144=81$$

Subtract 81 from both sides.

$$b^{2}+24b+144-81=0$$

Subtract 81 from 144 to get 63.

$$b^{2}+24b+63=0$$

To solve the equation, factor b^{2}+24b+63 using formula b^{2}+\left(a+b\right)b+ab=\left(b+a\right)\left(b+b\right). To find a and b, set up a system to be solved.

$$a+b=24 ab=63$$

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 63.

$$1,63 3,21 7,9$$

Calculate the sum for each pair.

$$1+63=64 3+21=24 7+9=16$$

The solution is the pair that gives sum 24.

$$a=3 b=21$$

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

$$\left(b+3\right)\left(b+21\right)$$

To find equation solutions, solve b+3=0 and b+21=0.

$$b=-3 b=-21$$

Steps Using Factoring By Grouping

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(b+12\right)^{2}.

$$b^{2}+24b+144=81$$

Subtract 81 from both sides.

$$b^{2}+24b+144-81=0$$

Subtract 81 from 144 to get 63.

$$b^{2}+24b+63=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as b^{2}+ab+bb+63. To find a and b, set up a system to be solved.

$$a+b=24 ab=1\times 63=63$$

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 63.

$$1,63 3,21 7,9$$

Calculate the sum for each pair.

$$1+63=64 3+21=24 7+9=16$$

The solution is the pair that gives sum 24.

$$a=3 b=21$$

Rewrite b^{2}+24b+63 as \left(b^{2}+3b\right)+\left(21b+63\right).

$$\left(b^{2}+3b\right)+\left(21b+63\right)$$

Factor out b in the first and 21 in the second group.

$$b\left(b+3\right)+21\left(b+3\right)$$

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

$$\left(b+3\right)\left(b+21\right)$$

To find equation solutions, solve b+3=0 and b+21=0.

$$b=-3 b=-21$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(b+12\right)^{2}.

$$b^{2}+24b+144=81$$

Subtract 81 from both sides.

$$b^{2}+24b+144-81=0$$

Subtract 81 from 144 to get 63.

$$b^{2}+24b+63=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 24 for b, and 63 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$b=\frac{-24±\sqrt{24^{2}-4\times 63}}{2}$$

Square 24.

$$b=\frac{-24±\sqrt{576-4\times 63}}{2}$$

Multiply -4 times 63.

$$b=\frac{-24±\sqrt{576-252}}{2}$$

Add 576 to -252.

$$b=\frac{-24±\sqrt{324}}{2}$$

Take the square root of 324.

$$b=\frac{-24±18}{2}$$

Now solve the equation b=\frac{-24±18}{2} when ± is plus. Add -24 to 18.

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

Divide -6 by 2.

$$b=-3$$

Now solve the equation b=\frac{-24±18}{2} when ± is minus. Subtract 18 from -24.

$$b=-\frac{42}{2}$$

Divide -42 by 2.

$$b=-21$$

The equation is now solved.

$$b=-3 b=-21$$

Steps for Completing the Square

Take the square root of both sides of the equation.

$$\sqrt{\left(b+12\right)^{2}}=\sqrt{81}$$

Simplify.

$$b+12=9 b+12=-9$$

Subtract 12 from both sides of the equation.

$$b=-3 b=-21$$