Multiple choice

If $c^2=4d$ and the two equations $x^2-ax+b=0$ and $x^2-cx+d=0$ have one common root, then the value of $2(b+d)$ is equal to

  1. $\displaystyle \frac{a}{c}$
  2. $ac$
  3. $2ac$
  4. $a+c$
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B Correct answer
Explanation

Let alpha be the common root. alpha^2 - a*alpha + b = 0 and alpha^2 - c*alpha + d = 0. Subtracting: (c-a)alpha + (b-d) = 0, so alpha = (d-b)/(c-a). Substituting back and using c^2=4d, the condition leads to 2(b+d) = ac.

AI explanation

Given c^2 = 4d, the discriminant of x^2 - cx + d = 0 is c^2 - 4d = 0, so its roots are equal at x = c/2. Let this be the common root, which means (c/2)^2 - a(c/2) + b = 0. Multiplying by 4 gives c^2 - 2ac + 4b = 0, and substituting 4d for c^2 results in 4b + 4d = 2ac, which simplifies to 2(b + d) = ac.

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