Solve x^2-3x-180

$${ x }^{ 2 } -3x-180$$

Factor

$$\left(x-15\right)\left(x+12\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx-180. To find a and b, set up a system to be solved.

$$a+b=-3 ab=1\left(-180\right)=-180$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. 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 -3.

$$a=-15 b=12$$

Rewrite x^{2}-3x-180 as \left(x^{2}-15x\right)+\left(12x-180\right).

$$\left(x^{2}-15x\right)+\left(12x-180\right)$$

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

$$x\left(x-15\right)+12\left(x-15\right)$$

Factor out common term x-15 by using distributive property.

$$\left(x-15\right)\left(x+12\right)$$

Steps Using the Quadratic Formula

Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

$$x^{2}-3x-180=0$$

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.

$$x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\left(-180\right)}}{2}$$

Square -3.

$$x=\frac{-\left(-3\right)±\sqrt{9-4\left(-180\right)}}{2}$$

Multiply -4 times -180.

$$x=\frac{-\left(-3\right)±\sqrt{9+720}}{2}$$

Add 9 to 720.

$$x=\frac{-\left(-3\right)±\sqrt{729}}{2}$$

Take the square root of 729.

$$x=\frac{-\left(-3\right)±27}{2}$$

The opposite of -3 is 3.

$$x=\frac{3±27}{2}$$

Now solve the equation x=\frac{3±27}{2} when ± is plus. Add 3 to 27.

$$x=\frac{30}{2}$$

Divide 30 by 2.

$$x=15$$

Now solve the equation x=\frac{3±27}{2} when ± is minus. Subtract 27 from 3.

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

Divide -24 by 2.

$$x=-12$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 15 for x_{1} and -12 for x_{2}.

$$x^{2}-3x-180=\left(x-15\right)\left(x-\left(-12\right)\right)$$

Simplify all the expressions of the form p-\left(-q\right) to p+q.

$$x^{2}-3x-180=\left(x-15\right)\left(x+12\right)$$

Evaluate

$$\left(x-15\right)\left(x+12\right)$$