Problem of the Week

PROBLEM 223

I received the following email from a colleague, Mary Jane Sterling, who is teaching a course in College Algebra this semester.  The student mentioned in the email will not be identified since he is a minor -- though one hopes not in mathematics. 

On Monday, I solved the equation √(3x + 7) + √(x + 2) = 1 by squaring both sides twice; the resulting quadratic equation yields the only solution: x = -2. A student came up after class and asked why he couldn't just add the terms under the radicals together first, giving the equation √(4x + 9) = 1.  I told him "that doesn't work" and showed him some very impressive (to me) examples using just numbers under the radicals.  He was unimpressed, because he said that his way works.  Yep!  The answer to his "revised version" is indeed also x = -2. 

Of course, we all know this was just a coincidence, but just how coincidental was it?  This week's question:

For which real values of a, b, c, d do the two equations

√(a x + b) + √(c x + d) = 1    and    √((a + c) x + (b + d)) = 1

have the same solutions? 

For those requiring greater precision:  What is the probability that the two equations above have the same solutions?

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ã2005 Alberto L. Delgado