Question

How do you factor the expression `x^2 - 51x + 144`?

Answer 1

`(x-3)(x-48)`

Explanation:

In order to factor this, we should look for two integers that meet the following conditions:

• Their sum is the coefficient of the middle term `mathbf((-51))`
• Their product is the coefficient of the final term `mathbf((144))`

From these criteria, we can already determine a couple characteristics of the two integers. Since their product is positive, both numbers are the same sign. Since the sum is negative, we know that both the numbers are negative.

To find the correct integer pair, we should look at all the factor pairs for `-144:`

• `-1,-144`
• `-2,-72`
• `color(red)(ul(color(black)(-3))),color(red)(ul(color(black)(-48)))`

Here, by working up, we can stop finding factor pairs since `-3+(-48)=-51` and `-3xx-48=144`.

This means `x^2-51x+144` can be factored as

`(xcolor(red)(ul(color(black)(-3))))(xcolor(red)(ul(color(black)(-48))))`

We can check this by redistributing the terms.

`(x-3)(x-48)=x^2-3x-48+144=x^2-51x+144`