Solve 144=left(x+1right)^2

$$144= { \left(x+1 \right) }^{ 2 }$$

Solve for x

$$x=11$$

$$x=-13$$

Steps Using Factoring

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+1\right)^{2}.

$$144=x^{2}+2x+1$$

Swap sides so that all variable terms are on the left hand side.

$$x^{2}+2x+1=144$$

Subtract 144 from both sides.

$$x^{2}+2x+1-144=0$$

Subtract 144 from 1 to get -143.

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

To solve the equation, factor x^{2}+2x-143 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.

$$a+b=2 ab=-143$$

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 -143.

$$-1,143 -11,13$$

Calculate the sum for each pair.

$$-1+143=142 -11+13=2$$

The solution is the pair that gives sum 2.

$$a=-11 b=13$$

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

$$\left(x-11\right)\left(x+13\right)$$

To find equation solutions, solve x-11=0 and x+13=0.

$$x=11 x=-13$$

Steps Using Factoring By Grouping

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+1\right)^{2}.

$$144=x^{2}+2x+1$$

Swap sides so that all variable terms are on the left hand side.

$$x^{2}+2x+1=144$$

Subtract 144 from both sides.

$$x^{2}+2x+1-144=0$$

Subtract 144 from 1 to get -143.

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

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-143. To find a and b, set up a system to be solved.

$$a+b=2 ab=1\left(-143\right)=-143$$

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 -143.

$$-1,143 -11,13$$

Calculate the sum for each pair.

$$-1+143=142 -11+13=2$$

The solution is the pair that gives sum 2.

$$a=-11 b=13$$

Rewrite x^{2}+2x-143 as \left(x^{2}-11x\right)+\left(13x-143\right).

$$\left(x^{2}-11x\right)+\left(13x-143\right)$$

Factor out x in the first and 13 in the second group.

$$x\left(x-11\right)+13\left(x-11\right)$$

Factor out common term x-11 by using distributive property.

$$\left(x-11\right)\left(x+13\right)$$

To find equation solutions, solve x-11=0 and x+13=0.

$$x=11 x=-13$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+1\right)^{2}.

$$144=x^{2}+2x+1$$

Swap sides so that all variable terms are on the left hand side.

$$x^{2}+2x+1=144$$

Subtract 144 from both sides.

$$x^{2}+2x+1-144=0$$

Subtract 144 from 1 to get -143.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 2 for b, and -143 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2±\sqrt{2^{2}-4\left(-143\right)}}{2}$$

Square 2.

$$x=\frac{-2±\sqrt{4-4\left(-143\right)}}{2}$$

Multiply -4 times -143.

$$x=\frac{-2±\sqrt{4+572}}{2}$$

Add 4 to 572.

$$x=\frac{-2±\sqrt{576}}{2}$$

Take the square root of 576.

$$x=\frac{-2±24}{2}$$

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

$$x=\frac{22}{2}$$

Divide 22 by 2.

$$x=11$$

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

$$x=-\frac{26}{2}$$

Divide -26 by 2.

$$x=-13$$

The equation is now solved.

$$x=11 x=-13$$

Steps for Completing the Square

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+1\right)^{2}.

$$144=x^{2}+2x+1$$

Swap sides so that all variable terms are on the left hand side.

$$x^{2}+2x+1=144$$

Factor x^{2}+2x+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+1\right)^{2}=144$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{144}$$

Simplify.

$$x+1=12 x+1=-12$$

Subtract 1 from both sides of the equation.

$$x=11 x=-13$$