Question

How to prove that √x±4 +√x-4=2√x-1

Answer 1

Given:

√x+4−√x−4=2

Squaring both sides, we get:

(x+4)−2√x+4√x−4+(x−4)

=4

That is:

2x−2√x2−16

=4

Divide both sides by 2

to get:

x−√x2−16=2

Add

√x2−16−2

to both sides to get:

x−2=√x2−16

Square both sides to get:

x2−4x+4=x2−16

Add

−x2+4x+16

to both sides to get:

20=4x

Transpose and divide both sides by

4

to get:

x=5

Since we have squared both sides of the equation - which not a reversible operation - we need to check that this solution we have reached is a solution of the original equation.

We find:

√(5)+4−√(5)−4=√9−√1=3−1=2

So

x=5 is a valid solution.