The solution of √5x-1+√x-1=2 is:

1955 AMC 12 Problem 12

Attempt Problem 12 of the 1955 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1955 AMC 12 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

12.

The solution of

\sqrt{5x-1}+\sqrt{x-1}=2

is:

x=2,

x=1

x=\dfrac23

x=2

x=1

x=0

The real-domain restriction

\sqrt{x-1}

gives

x\ge1

Test the simplest endpoint before squaring the equation

The domain requires

x\ge1.

x=1,

the left side is

2+0=2,

x=1

works. To see that there is no other solution, note that both radicals are nondecreasing for

x\ge1,

and

\sqrt{5x-1}

is strictly increasing. Therefore the sum exceeds

2

for every

x>1.

Thus, the correct answer is D.