Solve x^2-22x-75^

$${ x }^{ 2 } -22x- { 75 }^{ }$$

Factor

$$\left(x-25\right)\left(x+3\right)$$

Solution Steps

Multiply and combine like terms.

$$x^{2}-22x-75$$

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx-75. To find a and b, set up a system to be solved.

$$a+b=-22 ab=1\left(-75\right)=-75$$

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 -75.

$$1,-75 3,-25 5,-15$$

Calculate the sum for each pair.

$$1-75=-74 3-25=-22 5-15=-10$$

The solution is the pair that gives sum -22.

$$a=-25 b=3$$

Rewrite x^{2}-22x-75 as \left(x^{2}-25x\right)+\left(3x-75\right).

$$\left(x^{2}-25x\right)+\left(3x-75\right)$$

Factor out x in the first and 3 in the second group.

$$x\left(x-25\right)+3\left(x-25\right)$$

Factor out common term x-25 by using distributive property.

$$\left(x-25\right)\left(x+3\right)$$

Evaluate

$$\left(x-25\right)\left(x+3\right)$$

Solution Steps

Calculate 75 to the power of 1 and get 75.

$$x^{2}-22x-75$$