Question

How do you factor the trinomial `4x^2 + 20x + 25`?

Answer 1

`color(black)((2x+5))color(black)((2x+5))` or `(2x+5)^2`

Explanation:

`4x^2+20x+25`

I like to use the factor by grouping method when working with polynomials whose leading coefficient is not `1`.

We need to find two numbers that multiply to `100` (`4*25`) and that add to `20`:

`color(white)(..)+20`
`color(white)(..)xx100`
.......................
`color(white)(.)1xx100`
`color(white)(.)2xx50`
`color(white)(.)4xx25`
`color(white)(.)5xx20`
`color(white)()10xx10`

`10+10=20` and `10xx10=100`! It looks like we've found our two values. But, we're not done yet...

to factor by grouping, we still need to group. I'll set it up and walk you through it:
First, we take the leading value `(4x^2)` and the constant `(25)` and place them in two parentheses, jopined together with addition:
`(4x^2+ _ )+( _ +25)`
We need to leave spaces for our two numbers, `10` and `10`
Now we insert them into the blank spaces:

`(4x^2+10color(orange)(x))+(10color(orange)(x)+25)`
Note It's important to add that `x`. The system won't work without it!

Now we just factor the parenthases:
`color(red)(2x)color(blue)((2x+5))+color(red)(5)color(blue)((2x+5))`
if the two parenthases look the same, we did everyhtign right! Now comes the magic...

Our factors are `color(red)((2x+5))color(blue)((2x+5))`. That's the same thing as `(2x+5)^2`.