Question

Factor 4x^2+4x+1=0?

Answer 1

`(2x+1)^2 = 0`

Explanation:

Given:

`4x^2+4x+1 = 0`

Does the pattern of the coefficients look familiar?

Consider it in the form `441`

Did you know that `441 = 21^2`?

Similarly, we find:

`4x^2+4x+1 = (2x+1)^2`

Why does this work?

Think of putting `x = 10` to get:

`4(color(blue)(10))^2+4(color(blue)(10))+1 = 400+40+1 = 441`

`2(color(blue)(10))+1 = 20+1 = 21`

So it works because the coefficients are small enough that there are no carried digits when you square `21`.

We can similarly recognise:

`x^2+6x+9 = (x+3)^2" "` like `" "169=13^2`

`x^2+2x+1 = (x+1)^2" "` like `" "121 = 11^2`

Note that reversing the sign in the binomial has the effect of reversing the sign on the middle `x` term of the square, so we have:

`9x^2-6x+1 = (3x-1)^2" "` like `" "961 = 31^2`