Solve x^2+24x+119

$${ x }^{ 2 } +24x+119$$

Factor

$$\left(x+7\right)\left(x+17\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+119. To find a and b, set up a system to be solved.

$$a+b=24 ab=1\times 119=119$$

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 119.

$$1,119 7,17$$

Calculate the sum for each pair.

$$1+119=120 7+17=24$$

The solution is the pair that gives sum 24.

$$a=7 b=17$$

Rewrite x^{2}+24x+119 as \left(x^{2}+7x\right)+\left(17x+119\right).

$$\left(x^{2}+7x\right)+\left(17x+119\right)$$

Factor out x in the first and 17 in the second group.

$$x\left(x+7\right)+17\left(x+7\right)$$

Factor out common term x+7 by using distributive property.

$$\left(x+7\right)\left(x+17\right)$$

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.

$$x^{2}+24x+119=0$$

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.

$$x=\frac{-24±\sqrt{24^{2}-4\times 119}}{2}$$

Square 24.

$$x=\frac{-24±\sqrt{576-4\times 119}}{2}$$

Multiply -4 times 119.

$$x=\frac{-24±\sqrt{576-476}}{2}$$

Add 576 to -476.

$$x=\frac{-24±\sqrt{100}}{2}$$

Take the square root of 100.

$$x=\frac{-24±10}{2}$$

Now solve the equation x=\frac{-24±10}{2} when ± is plus. Add -24 to 10.

$$x=-\frac{14}{2}$$

Divide -14 by 2.

$$x=-7$$

Now solve the equation x=\frac{-24±10}{2} when ± is minus. Subtract 10 from -24.

$$x=-\frac{34}{2}$$

Divide -34 by 2.

$$x=-17$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -7 for x_{1} and -17 for x_{2}.

$$x^{2}+24x+119=\left(x-\left(-7\right)\right)\left(x-\left(-17\right)\right)$$

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

$$x^{2}+24x+119=\left(x+7\right)\left(x+17\right)$$

Evaluate

$$\left(x+7\right)\left(x+17\right)$$