Solve x^2+24x=-144

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

Solve for x

$$x=-12$$

Steps Using Factoring

Add 144 to both sides.

$$x^{2}+24x+144=0$$

To solve the equation, factor x^{2}+24x+144 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.

$$a+b=24 ab=144$$

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 144.

$$1,144 2,72 3,48 4,36 6,24 8,18 9,16 12,12$$

Calculate the sum for each pair.

$$1+144=145 2+72=74 3+48=51 4+36=40 6+24=30 8+18=26 9+16=25 12+12=24$$

The solution is the pair that gives sum 24.

$$a=12 b=12$$

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

$$\left(x+12\right)\left(x+12\right)$$

Rewrite as a binomial square.

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

To find equation solution, solve x+12=0.

$$x=-12$$

Steps Using Factoring By Grouping

Add 144 to both sides.

$$x^{2}+24x+144=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+144. To find a and b, set up a system to be solved.

$$a+b=24 ab=1\times 144=144$$

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 144.

$$1,144 2,72 3,48 4,36 6,24 8,18 9,16 12,12$$

Calculate the sum for each pair.

$$1+144=145 2+72=74 3+48=51 4+36=40 6+24=30 8+18=26 9+16=25 12+12=24$$

The solution is the pair that gives sum 24.

$$a=12 b=12$$

Rewrite x^{2}+24x+144 as \left(x^{2}+12x\right)+\left(12x+144\right).

$$\left(x^{2}+12x\right)+\left(12x+144\right)$$

Factor out x in the first and 12 in the second group.

$$x\left(x+12\right)+12\left(x+12\right)$$

Factor out common term x+12 by using distributive property.

$$\left(x+12\right)\left(x+12\right)$$

Rewrite as a binomial square.

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

To find equation solution, solve x+12=0.

$$x=-12$$

Steps Using the Quadratic Formula

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

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

Add 144 to both sides of the equation.

$$x^{2}+24x-\left(-144\right)=-144-\left(-144\right)$$

Subtracting -144 from itself leaves 0.

$$x^{2}+24x-\left(-144\right)=0$$

Subtract -144 from 0.

$$x^{2}+24x+144=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 24 for b, and 144 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-24±\sqrt{24^{2}-4\times 144}}{2}$$

Square 24.

$$x=\frac{-24±\sqrt{576-4\times 144}}{2}$$

Multiply -4 times 144.

$$x=\frac{-24±\sqrt{576-576}}{2}$$

Add 576 to -576.

$$x=\frac{-24±\sqrt{0}}{2}$$

Take the square root of 0.

$$x=-\frac{24}{2}$$

Divide -24 by 2.

$$x=-12$$

Steps for Completing the Square

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

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

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.

$$x^{2}+24x+12^{2}=-144+12^{2}$$

Square 12.

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

Add -144 to 144.

$$x^{2}+24x+144=0$$

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}.

$$\left(x+12\right)^{2}=0$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+12\right)^{2}}=\sqrt{0}$$

Simplify.

$$x+12=0 x+12=0$$

Subtract 12 from both sides of the equation.

$$x=-12 x=-12$$

The equation is now solved. Solutions are the same.

$$x=-12$$