Solve x^2+24x+0=180

$$x ^ { 2 } + 24 x + 0 = 180$$

Solve for x

$$x=-30$$

$$x=6$$

Steps Using Factoring

Anything plus zero gives itself.

$$x^{2}+24x=180$$

Subtract 180 from both sides.

$$x^{2}+24x-180=0$$

To solve the equation, factor x^{2}+24x-180 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.

$$a+b=24 ab=-180$$

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 -180.

$$-1,180 -2,90 -3,60 -4,45 -5,36 -6,30 -9,20 -10,18 -12,15$$

Calculate the sum for each pair.

$$-1+180=179 -2+90=88 -3+60=57 -4+45=41 -5+36=31 -6+30=24 -9+20=11 -10+18=8 -12+15=3$$

The solution is the pair that gives sum 24.

$$a=-6 b=30$$

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

$$\left(x-6\right)\left(x+30\right)$$

To find equation solutions, solve x-6=0 and x+30=0.

$$x=6 x=-30$$

Steps Using Factoring By Grouping

Anything plus zero gives itself.

$$x^{2}+24x=180$$

Subtract 180 from both sides.

$$x^{2}+24x-180=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-180. To find a and b, set up a system to be solved.

$$a+b=24 ab=1\left(-180\right)=-180$$

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 -180.

$$-1,180 -2,90 -3,60 -4,45 -5,36 -6,30 -9,20 -10,18 -12,15$$

Calculate the sum for each pair.

$$-1+180=179 -2+90=88 -3+60=57 -4+45=41 -5+36=31 -6+30=24 -9+20=11 -10+18=8 -12+15=3$$

The solution is the pair that gives sum 24.

$$a=-6 b=30$$

Rewrite x^{2}+24x-180 as \left(x^{2}-6x\right)+\left(30x-180\right).

$$\left(x^{2}-6x\right)+\left(30x-180\right)$$

Factor out x in the first and 30 in the second group.

$$x\left(x-6\right)+30\left(x-6\right)$$

Factor out common term x-6 by using distributive property.

$$\left(x-6\right)\left(x+30\right)$$

To find equation solutions, solve x-6=0 and x+30=0.

$$x=6 x=-30$$

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.

$$x^{2}+24x=180$$

Subtract 180 from both sides of the equation.

$$x^{2}+24x-180=180-180$$

Subtracting 180 from itself leaves 0.

$$x^{2}+24x-180=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 24 for b, and -180 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-24±\sqrt{24^{2}-4\left(-180\right)}}{2}$$

Square 24.

$$x=\frac{-24±\sqrt{576-4\left(-180\right)}}{2}$$

Multiply -4 times -180.

$$x=\frac{-24±\sqrt{576+720}}{2}$$

Add 576 to 720.

$$x=\frac{-24±\sqrt{1296}}{2}$$

Take the square root of 1296.

$$x=\frac{-24±36}{2}$$

Now solve the equation x=\frac{-24±36}{2} when ± is plus. Add -24 to 36.

$$x=\frac{12}{2}$$

Divide 12 by 2.

$$x=6$$

Now solve the equation x=\frac{-24±36}{2} when ± is minus. Subtract 36 from -24.

$$x=-\frac{60}{2}$$

Divide -60 by 2.

$$x=-30$$

The equation is now solved.

$$x=6 x=-30$$

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.

$$x^{2}+24x=180$$

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.

$$x^{2}+24x+12^{2}=180+12^{2}$$

Square 12.

$$x^{2}+24x+144=180+144$$

Add 180 to 144.

$$x^{2}+24x+144=324$$

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}.

$$\left(x+12\right)^{2}=324$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+12\right)^{2}}=\sqrt{324}$$

Simplify.

$$x+12=18 x+12=-18$$

Subtract 12 from both sides of the equation.

$$x=6 x=-30$$