Question

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

Answer 1

(x - 10)(x - 12)

Explanation:

`y = x^2 - 22x + 120.`

Standard form `y=ax^2+bx+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)

Answer 2

Using reasoning:

`x^2-22x+120" "=" "(x-10)(x-12)`

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)->`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^2-22x+120" "=" "(x-10)(x-12)`