Multiple choice

If the equations x2 – px + q = 0 and x2 – ax + b = 0 have a common root and the roots of the second equation are equal, then which of the following is correct?

  1. aq = 2(b + p)

  2. aq = b + p

  3. ap = 2(b + q)

  4. ap = b + q
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Roots of x^2 - ax + b = 0 are equal, so discriminant = a^2 - 4b = 0 => a^2 = 4b. Let the common root be k. k^2 - pk + q = 0 and k^2 - ak + b = 0. Subtracting: (a-p)k = b-q => k = (b-q)/(a-p). Substituting back into the first equation and using a^2=4b leads to the relation ap = 2(b+q).

AI explanation

Since the roots of x^2 - ax + b = 0 are equal, its discriminant is zero, so a^2 = 4b, and the repeated common root is a/2. Substituting x = a/2 into the first equation x^2 - px + q = 0 gives (a/2)^2 - p(a/2) + q = 0. Multiplying by 4 yields a^2 - 2ap + 4q = 0. Replacing a^2 with 4b from the previous discriminant condition gives 4b - 2ap + 4q = 0, which rearranges to ap = 2(b + q).

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