Solve x^2-14x+24

$${ x }^{ 2 } -14x+24$$

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.

$$a+b=-14 ab=1\times 24=24$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 24.

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

Calculate the sum for each pair.

$$-1-24=-25 -2-12=-14 -3-8=-11 -4-6=-10$$

The solution is the pair that gives sum -14.

$$a=-12 b=-2$$

Rewrite x^{2}-14x+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.

$$x^{2}-14x+24=0$$

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.

$$x=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\times 24}}{2}$$

Square -14.

$$x=\frac{-\left(-14\right)±\sqrt{196-4\times 24}}{2}$$

Multiply -4 times 24.

$$x=\frac{-\left(-14\right)±\sqrt{196-96}}{2}$$

Add 196 to -96.

$$x=\frac{-\left(-14\right)±\sqrt{100}}{2}$$

Take the square root of 100.

$$x=\frac{-\left(-14\right)±10}{2}$$

The opposite of -14 is 14.

$$x=\frac{14±10}{2}$$

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

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

Divide 24 by 2.

$$x=12$$

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

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

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}.

$$x^{2}-14x+24=\left(x-12\right)\left(x-2\right)$$

Evaluate

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