Question

How do you solve `-4 sqrt(x+9) = 20` and find any extraneous solutions?

Answer 1

`x in O/`

Explanation:

Your radical equation has no solutions for real numbers. Here's why.

Isolate the square root on one side of the equation by dividing both sides by `-4`

`(color(red)(cancel(color(black)(-4))) * sqrt(x+9))/color(red)(cancel(color(black)(-4))) = 20/(-4)`

`sqrt(x+9) = -5`

By definition, the square root of a positive real number must always produce a positive value. This implies that you must have

• `x + 9 >= 0 ->` you must take the square root of a positive number when working in `RR`

• `color(red)(cancel(color(black)(-5 >=0))) ->` the square root of a positive number must be a positive number

In your case ,`-5` is clearly not positive, which implies that your equation has no solutions when working in `RR`. Simply put, you can't find a value of `x in RR` that can satisfy both conditions.

You can have `x+9 >=0` for `x in [-9, +oo)`, but you will never have `-5 >=0`, regardless of the value of `x in [-9, + oo)`.

You can write this as

`x in O/ ->` no solutions in `RR`