Solve (x+12)^2-1=80

$$( x + 12 ) ^ { 2 } - 1 = 80$$

Solve for x

$$x=-3$$

$$x=-21$$

Steps Using Factoring

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

$$x^{2}+24x+144-1=80$$

Subtract 1 from 144 to get 143.

$$x^{2}+24x+143=80$$

Subtract 80 from both sides.

$$x^{2}+24x+143-80=0$$

Subtract 80 from 143 to get 63.

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

To solve the equation, factor x^{2}+24x+63 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.

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

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

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

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

$$x=-3 x=-21$$

Steps Using Factoring By Grouping

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

$$x^{2}+24x+144-1=80$$

Subtract 1 from 144 to get 143.

$$x^{2}+24x+143=80$$

Subtract 80 from both sides.

$$x^{2}+24x+143-80=0$$

Subtract 80 from 143 to get 63.

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

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+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 x^{2}+24x+63 as \left(x^{2}+3x\right)+\left(21x+63\right).

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

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

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

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

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

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

$$x=-3 x=-21$$

Steps Using the Quadratic Formula

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

$$x^{2}+24x+144-1=80$$

Subtract 1 from 144 to get 143.

$$x^{2}+24x+143=80$$

Subtract 80 from both sides.

$$x^{2}+24x+143-80=0$$

Subtract 80 from 143 to get 63.

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

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}.

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

Square 24.

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

Multiply -4 times 63.

$$x=\frac{-24±\sqrt{576-252}}{2}$$

Add 576 to -252.

$$x=\frac{-24±\sqrt{324}}{2}$$

Take the square root of 324.

$$x=\frac{-24±18}{2}$$

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

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

Divide -6 by 2.

$$x=-3$$

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

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

Divide -42 by 2.

$$x=-21$$

The equation is now solved.

$$x=-3 x=-21$$

Steps for Completing the Square

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

$$x^{2}+24x+144-1=80$$

Subtract 1 from 144 to get 143.

$$x^{2}+24x+143=80$$

Subtract 143 from both sides.

$$x^{2}+24x=80-143$$

Subtract 143 from 80 to get -63.

$$x^{2}+24x=-63$$

Divide 24, the coefficient of the x term, by 2 to get 12. Then add the square of 12 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+24x+12^{2}=-63+12^{2}$$

Square 12.

$$x^{2}+24x+144=-63+144$$

Add -63 to 144.

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

Factor x^{2}+24x+144. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$x+12=9 x+12=-9$$

Subtract 12 from both sides of the equation.

$$x=-3 x=-21$$