Solve x^2-10x-24

$${ x }^{ 2 } -10x-24$$

Factor

$$\left(x-12\right)\left(x+2\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx-24. To find a and b, set up a system to be solved.

$$a+b=-10 ab=1\left(-24\right)=-24$$

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 -24.

$$1,-24 2,-12 3,-8 4,-6$$

Calculate the sum for each pair.

$$1-24=-23 2-12=-10 3-8=-5 4-6=-2$$

The solution is the pair that gives sum -10.

$$a=-12 b=2$$

Rewrite x^{2}-10x-24 as \left(x^{2}-12x\right)+\left(2x-24\right).

$$\left(x^{2}-12x\right)+\left(2x-24\right)$$

Factor out x in the first and 2 in the second group.

$$x\left(x-12\right)+2\left(x-12\right)$$

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

$$\left(x-12\right)\left(x+2\right)$$

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.

$$x^{2}-10x-24=0$$

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.

$$x=\frac{-\left(-10\right)±\sqrt{\left(-10\right)^{2}-4\left(-24\right)}}{2}$$

Square -10.

$$x=\frac{-\left(-10\right)±\sqrt{100-4\left(-24\right)}}{2}$$

Multiply -4 times -24.

$$x=\frac{-\left(-10\right)±\sqrt{100+96}}{2}$$

Add 100 to 96.

$$x=\frac{-\left(-10\right)±\sqrt{196}}{2}$$

Take the square root of 196.

$$x=\frac{-\left(-10\right)±14}{2}$$

The opposite of -10 is 10.

$$x=\frac{10±14}{2}$$

Now solve the equation x=\frac{10±14}{2} when ± is plus. Add 10 to 14.

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

Divide 24 by 2.

$$x=12$$

Now solve the equation x=\frac{10±14}{2} when ± is minus. Subtract 14 from 10.

$$x=-\frac{4}{2}$$

Divide -4 by 2.

$$x=-2$$

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

$$x^{2}-10x-24=\left(x-12\right)\left(x-\left(-2\right)\right)$$

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

$$x^{2}-10x-24=\left(x-12\right)\left(x+2\right)$$

Evaluate

$$\left(x-12\right)\left(x+2\right)$$