Question

How do you solve `4(x+1)^2=100`?

Answer 1

`x = 4" "` or `" "x = -6`

Explanation:

The key to this problem is the fact that for a given number `a in RR`, you have

`a^2 = a * a" "` and `" "a^2 = (-a) * (-a)`

so right from the start you know that you're looking for two values of `x`, since for `a in RR`, you have

`color(purple)(|bar(ul(color(white)(a/a)color(black)(sqrt(a^2) = |a| = {(+ a, " for "a >=0), (-a, " for "a < 0):})color(white)(a/a)|)))`

The first thing to do here is divide both sides of the equation by `4`

`(color(red)(cancel(color(black)(4))) * (x+1)^2)/color(red)(cancel(color(black)(4))) = 100/4`

`(x+1)^2 = 25`

If you take the square root of both sides of the equation

`sqrt((x+1)^2) = sqrt(25)`

you will end up with

`(x+1) = + 5" "` or `" "(x+1) = -5`

This will get you

`x+1 = 5 implies x= 4" "` or `" " x+1 = -5 implies x= -6`

Therefore, the original equation has two possible solutions

`x = 4" "` or `" "x = -6`

Do a quick check to make sure that the calculations are correct

`x = 4:" " 4 * (4 + 1)^2 = 100`

`4 * 5^2 = 100" "color(green)(sqrt())`

`x = -6:" " 4 * (-6 + 1)^2 = 100`

`4 * (-5)^2 = 100" "color(green)(sqrt())`