Solve x^2+19x+88

$${ x }^{ 2 } +19x+88$$

Factor

$$\left(x+8\right)\left(x+11\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+88. To find a and b, set up a system to be solved.

$$a+b=19 ab=1\times 88=88$$

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 88.

$$1,88 2,44 4,22 8,11$$

Calculate the sum for each pair.

$$1+88=89 2+44=46 4+22=26 8+11=19$$

The solution is the pair that gives sum 19.

$$a=8 b=11$$

Rewrite x^{2}+19x+88 as \left(x^{2}+8x\right)+\left(11x+88\right).

$$\left(x^{2}+8x\right)+\left(11x+88\right)$$

Factor out x in the first and 11 in the second group.

$$x\left(x+8\right)+11\left(x+8\right)$$

Factor out common term x+8 by using distributive property.

$$\left(x+8\right)\left(x+11\right)$$

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.

$$x^{2}+19x+88=0$$

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.

$$x=\frac{-19±\sqrt{19^{2}-4\times 88}}{2}$$

Square 19.

$$x=\frac{-19±\sqrt{361-4\times 88}}{2}$$

Multiply -4 times 88.

$$x=\frac{-19±\sqrt{361-352}}{2}$$

Add 361 to -352.

$$x=\frac{-19±\sqrt{9}}{2}$$

Take the square root of 9.

$$x=\frac{-19±3}{2}$$

Now solve the equation x=\frac{-19±3}{2} when ± is plus. Add -19 to 3.

$$x=-\frac{16}{2}$$

Divide -16 by 2.

$$x=-8$$

Now solve the equation x=\frac{-19±3}{2} when ± is minus. Subtract 3 from -19.

$$x=-\frac{22}{2}$$

Divide -22 by 2.

$$x=-11$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -8 for x_{1} and -11 for x_{2}.

$$x^{2}+19x+88=\left(x-\left(-8\right)\right)\left(x-\left(-11\right)\right)$$

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

$$x^{2}+19x+88=\left(x+8\right)\left(x+11\right)$$

Evaluate

$$\left(x+8\right)\left(x+11\right)$$