Question

How do you solve `3 - x =sqrt(x^2 + 15)`?

Answer 1

`x = -1`

Explanation:

Start by taking a look at the expression that's under the square root.

`x^2 + 15>0, (AA) x in RR`

In other words, as far as the radical term is concerned, `x` can take any value in `RR`. On the other hand, the possible values of `x` are restricted by the expression on the left side of the equation.

Since the square root of a positive number is always a positive number, you have

• `3-x>=0 implies x<=3`

Now, square both sides of the equation to get rid of the radical term

`(3-x)^2 = (sqrt(x^2 + 15))^2`

`9 - 6x + color(red)(cancel(color(black)(x^2))) = color(red)(cancel(color(black)(x^2))) + 15`

This is equivalent to

`-6x = 6 implies x= 6/(-6) = color(green)(-1)`

Since `x=-1` satisfies the condition `x<=3`, this will be the solution to the original equation.

Do a quick check to make sure that everything came out right

`3 - (-1) = sqrt( (-1)^2 + 15)`

`3 + 1 = sqrt(16)`

`4 = 4 color(green)(sqrt())`