How to check the irrational solution to a rational equation?

Question

After solving a rational equation leading to two irrational solutions, if you have to algebraically confirm those solutions, how do you do so?

The rational equation:

x/(x+2) + 2/x = -5/(2x+4)

The solutions are:

(-9 ±√17)/4

Let's say you had to algebraically show that (-9+√17)/4 is in fact a solution. How...

Answer 1

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Answer 2

If (-9 ±√17)/4 is a solution for x/(x+2) + 2/x = -5/(2x+4), then you would simply need to show that when you plug in (-9 +√17)/4 and (-9 -√17)/4 for x, the simplified expression holds true. To make this a bit easier you can simplify x/(x+2) + 2/x = -5/(2x+4) further.

Summing x/(x+2) and 2/x gives you:

(x^2+2x+4)/(x*(x+2)) = -5/(2x+4)

You can cancel out the x+2 in the denominator by factoring out 2 from 2x+4. Your simplified expression would be:

(x^2+2x+4)/x = -2.5

Now you would plug in (-9 +√17)/4 and (-9 -√17)/4 into the simplified expression to show that it equals -2.5 on both sides. Hope this helps!