How to Solve 4\sqrt{x-2} + 8=3\sqrt{x+6}-5 for x | Refresh Your Memory

[DB]

my teacher showed me how to solve this but she said we don't have to know it till cegep. id still like to know how to do it, can u guys help me refresh my memory?

$$-4\sqrt{x-2} + 8=3\sqrt{x+6}-5$$

solve for x

 

[TD]

You can solve these kind of irrational equations by squaring both sides, it may be necessary to do so more than once.
You square because you want to get rid off the square roots, so it's necessary to carefully choose what to put on what side because you don't want to create new square roots as a result of mixed terms, or at least you want to minimize the number of those.

You have to be careful though, when squaring - the sign of both sides has to be the same and all expressions under the square roots can't be negative, these constraints give you conditions to test on your final answers - some may have to be cancelled.

 

[Architeuthis Dux]

To solve this equation, we need to isolate the variable x on one side of the equation. We can do this by moving all the terms with x to one side and all the constant terms to the other side. Let's start by moving the constant terms to the right side of the equation:

-4\sqrt{x-2} + 8=3\sqrt{x+6}-5
-4\sqrt{x-2} = 3\sqrt{x+6}-5-8
-4\sqrt{x-2} = 3\sqrt{x+6}-13

Next, we can square both sides of the equation to eliminate the square root:

(-4\sqrt{x-2})^2 = (3\sqrt{x+6}-13)^2
16(x-2) = 9(x+6)-78
16x-32 = 9x+54-78
16x-9x = 54-78+32
7x = 8
x = 8/7

Therefore, the solution to the equation is x = 8/7. We can check this by plugging in the value of x back into the original equation:

-4\sqrt{(8/7)-2} + 8 = 3\sqrt{(8/7)+6}-5
-4\sqrt{(8/7)-14/7} + 8 = 3\sqrt{(8/7)+42/7}-5
-4\sqrt{-6/7} + 8 = 3\sqrt{50/7}-5
-4\sqrt{-6/7} + 8 = 3(5\sqrt{2}/7)-5
-4(-6/7) + 8 = 15\sqrt{2}/7-5
24/7 + 8 = 15\sqrt{2}/7-5
(24+56)/7 = (15\sqrt{2}-35)/7
80/7 = 80/7

Therefore, our solution of x = 8/7 is correct. This is one method of solving the equation, but there may be other methods that your teacher will introduce in the future. It's always good to keep an open mind and be willing to learn new techniques as you continue your studies in science.