# How do you factor x^2 - 22x + 120?

Mar 25, 2016

(x - 10)(x - 12)

#### Explanation:

$y = {x}^{2} - 22 x + 120.$

Standard form $y = a {x}^{2} + b x + c$

Use the new Transforming Method (Google, Yahoo)
Find 2 numbers knowing sum (- 22 = b) and product (ac = 120). The numbers have same sign because ac > 0.
Factor pairs of (120) --> ...(6, 20)(10, 12). This last sum is 22 = -b. Then the opposite sum (-10, -12) gives the 2 numbers: -10 and -12 (sum = b)
y = (x - 10)(x - 12)

May 22, 2017

Using reasoning:

${x}^{2} - 22 x + 120 \text{ "=" } \left(x - 10\right) \left(x - 12\right)$

#### Explanation:

Lets try and reason this out.

The coefficient of ${x}^{2}$ is +1 so as a starting point we have the form

( x+-?_1)(x+-?_2)

The constant of 120 is positive So both of the ? are the same sign.

The $x$ term has negative 22 so both ? are negative giving

(x-?_1)(x-?_2)

Note that $\sqrt{120} \to$just under 11

Stating the obvious, the constant of 120 ends in zero so it has to be the product of two numbers where one of them ends in either 5 or 0.

As the square root is close to 11 lets consider the possibility one of the factors as being 10.

10xx12=120 and (-10)+(-12) = -22" " larrcolor(red)(" It works")

So we have:

${x}^{2} - 22 x + 120 \text{ "=" } \left(x - 10\right) \left(x - 12\right)$