Multiple choice

If the two equations $ x^{2}-cx+d=0 $ and $ x^{2}-ax+b=0 $ have a common root and the second equation has equal roots, then

  1. $b+d = ac$
  2. $2 (b+d) = ac$
  3. $b+d=2ac$
  4. $ (b+d)^{2} =a+c $
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B Correct answer
Explanation

Eq 2: x^2 - ax + b = 0 has equal roots, so discriminant = a^2 - 4b = 0, meaning b = a^2/4. Let alpha be the common root. alpha^2 - c*alpha + d = 0 and alpha^2 - a*alpha + b = 0. Subtracting: (a-c)alpha + (d-b) = 0, so alpha = (b-d)/(a-c). Substituting back into x^2 - ax + b = 0: ((b-d)/(a-c))^2 - a((b-d)/(a-c)) + b = 0. This simplifies to 2(b+d) = ac.

AI explanation

Since x^2 - ax + b = 0 has equal roots, its discriminant is zero, giving a^2 = 4b, and the common root is x = a/2. Substituting x = a/2 into the first equation x^2 - cx + d = 0 gives a^2/4 - ac/2 + d = 0. Multiplying by 4 and substituting 4b for a^2 results in 2(b + d) = ac.

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