Solve x^2+7x-44

$${ x }^{ 2 } +7x-44$$

Factor

$$\left(x-4\right)\left(x+11\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx-44. To find a and b, set up a system to be solved.

$$a+b=7 ab=1\left(-44\right)=-44$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -44.

$$-1,44 -2,22 -4,11$$

Calculate the sum for each pair.

$$-1+44=43 -2+22=20 -4+11=7$$

The solution is the pair that gives sum 7.

$$a=-4 b=11$$

Rewrite x^{2}+7x-44 as \left(x^{2}-4x\right)+\left(11x-44\right).

$$\left(x^{2}-4x\right)+\left(11x-44\right)$$

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

$$x\left(x-4\right)+11\left(x-4\right)$$

Factor out common term x-4 by using distributive property.

$$\left(x-4\right)\left(x+11\right)$$

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.

$$x^{2}+7x-44=0$$

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.

$$x=\frac{-7±\sqrt{7^{2}-4\left(-44\right)}}{2}$$

Square 7.

$$x=\frac{-7±\sqrt{49-4\left(-44\right)}}{2}$$

Multiply -4 times -44.

$$x=\frac{-7±\sqrt{49+176}}{2}$$

Add 49 to 176.

$$x=\frac{-7±\sqrt{225}}{2}$$

Take the square root of 225.

$$x=\frac{-7±15}{2}$$

Now solve the equation x=\frac{-7±15}{2} when ± is plus. Add -7 to 15.

$$x=\frac{8}{2}$$

Divide 8 by 2.

$$x=4$$

Now solve the equation x=\frac{-7±15}{2} when ± is minus. Subtract 15 from -7.

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

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 4 for x_{1} and -11 for x_{2}.

$$x^{2}+7x-44=\left(x-4\right)\left(x-\left(-11\right)\right)$$

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

$$x^{2}+7x-44=\left(x-4\right)\left(x+11\right)$$

Evaluate

$$\left(x-4\right)\left(x+11\right)$$