Solve x^2-22x-968

$${ x }^{ 2 } -22x-968$$

Factor

$$\left(x-44\right)\left(x+22\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx-968. To find a and b, set up a system to be solved.

$$a+b=-22 ab=1\left(-968\right)=-968$$

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 -968.

$$1,-968 2,-484 4,-242 8,-121 11,-88 22,-44$$

Calculate the sum for each pair.

$$1-968=-967 2-484=-482 4-242=-238 8-121=-113 11-88=-77 22-44=-22$$

The solution is the pair that gives sum -22.

$$a=-44 b=22$$

Rewrite x^{2}-22x-968 as \left(x^{2}-44x\right)+\left(22x-968\right).

$$\left(x^{2}-44x\right)+\left(22x-968\right)$$

Factor out x in the first and 22 in the second group.

$$x\left(x-44\right)+22\left(x-44\right)$$

Factor out common term x-44 by using distributive property.

$$\left(x-44\right)\left(x+22\right)$$

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.

$$x^{2}-22x-968=0$$

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.

$$x=\frac{-\left(-22\right)±\sqrt{\left(-22\right)^{2}-4\left(-968\right)}}{2}$$

Square -22.

$$x=\frac{-\left(-22\right)±\sqrt{484-4\left(-968\right)}}{2}$$

Multiply -4 times -968.

$$x=\frac{-\left(-22\right)±\sqrt{484+3872}}{2}$$

Add 484 to 3872.

$$x=\frac{-\left(-22\right)±\sqrt{4356}}{2}$$

Take the square root of 4356.

$$x=\frac{-\left(-22\right)±66}{2}$$

The opposite of -22 is 22.

$$x=\frac{22±66}{2}$$

Now solve the equation x=\frac{22±66}{2} when ± is plus. Add 22 to 66.

$$x=\frac{88}{2}$$

Divide 88 by 2.

$$x=44$$

Now solve the equation x=\frac{22±66}{2} when ± is minus. Subtract 66 from 22.

$$x=-\frac{44}{2}$$

Divide -44 by 2.

$$x=-22$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 44 for x_{1} and -22 for x_{2}.

$$x^{2}-22x-968=\left(x-44\right)\left(x-\left(-22\right)\right)$$

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

$$x^{2}-22x-968=\left(x-44\right)\left(x+22\right)$$

Evaluate

$$\left(x-44\right)\left(x+22\right)$$