How do you solve # sqrt[x-5]-sqrt[x-3]=4# and find any extraneous solutions?

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Answer 1 · 2017-08-31T02:34:02

GIven: #sqrt(x-5)-sqrt(x-3)=4#

If you multiply both sides by the conjugate, #(sqrt(x-5)+sqrt(x-3))#, the radicals on the left disappear:

#(x-5)-(x-3)=4(sqrt(x-5)+sqrt(x-3))#

Please observe that the left side simplifies to a negative number:

#-2 = 4(sqrt(x-5)+sqrt(x-3))#

Divide both sides by 4 and flip the equation:

#sqrt(x-5)+sqrt(x-3) = -1/2#

This means that no solution exists.

Think of it this way.

You are starting with

#sqrt(x-5)#

(which must be a positive number) and you are subtracting another positive number

#sqrt(x-3)#

and you obtain a positive number

4

but, if you add the two numbers, you obtain a negative number, -1/2?

No.