Solve x^2-50x+96

$${ x }^{ 2 } -50x+96$$

Factor

$$\left(x-48\right)\left(x-2\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+96. To find a and b, set up a system to be solved.

$$a+b=-50 ab=1\times 96=96$$

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 96.

$$-1,-96 -2,-48 -3,-32 -4,-24 -6,-16 -8,-12$$

Calculate the sum for each pair.

$$-1-96=-97 -2-48=-50 -3-32=-35 -4-24=-28 -6-16=-22 -8-12=-20$$

The solution is the pair that gives sum -50.

$$a=-48 b=-2$$

Rewrite x^{2}-50x+96 as \left(x^{2}-48x\right)+\left(-2x+96\right).

$$\left(x^{2}-48x\right)+\left(-2x+96\right)$$

Factor out x in the first and -2 in the second group.

$$x\left(x-48\right)-2\left(x-48\right)$$

Factor out common term x-48 by using distributive property.

$$\left(x-48\right)\left(x-2\right)$$

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.

$$x^{2}-50x+96=0$$

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.

$$x=\frac{-\left(-50\right)±\sqrt{\left(-50\right)^{2}-4\times 96}}{2}$$

Square -50.

$$x=\frac{-\left(-50\right)±\sqrt{2500-4\times 96}}{2}$$

Multiply -4 times 96.

$$x=\frac{-\left(-50\right)±\sqrt{2500-384}}{2}$$

Add 2500 to -384.

$$x=\frac{-\left(-50\right)±\sqrt{2116}}{2}$$

Take the square root of 2116.

$$x=\frac{-\left(-50\right)±46}{2}$$

The opposite of -50 is 50.

$$x=\frac{50±46}{2}$$

Now solve the equation x=\frac{50±46}{2} when ± is plus. Add 50 to 46.

$$x=\frac{96}{2}$$

Divide 96 by 2.

$$x=48$$

Now solve the equation x=\frac{50±46}{2} when ± is minus. Subtract 46 from 50.

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

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 48 for x_{1} and 2 for x_{2}.

$$x^{2}-50x+96=\left(x-48\right)\left(x-2\right)$$

Evaluate

$$\left(x-48\right)\left(x-2\right)$$