Multiple choice

If the equations $\displaystyle 2x^{2}-7x+1=0$ and $ax^{2}+bx+2=0$ ($a ,b $ are rational) have a common root, then

  1. $a=2,b=-7$
  2. $\displaystyle a=-\frac{7}{2},b=1$
  3. $a=4,b=-14$

  4. none of these
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C Correct answer
Explanation

If they have a common root alpha, then 2*alpha^2 - 7*alpha + 1 = 0 and a*alpha^2 + b*alpha + 2 = 0. Multiplying the first by 2 gives 4*alpha^2 - 14*alpha + 2 = 0. Comparing coefficients with the second equation, we get a=4 and b=-14.

AI explanation

The discriminant of the first equation 2x^2 - 7x + 1 = 0 is (-7)^2 - 4(2)(1) = 41, which is not a perfect square, so its roots are irrational. Because a and b are rational in the second equation ax^2 + bx + 2 = 0, for them to share a common root, the second equation must be a direct multiple of the first. Multiplying the first equation by 2 gives 4x^2 - 14x + 2 = 0. Matching the coefficients reveals that a = 4 and b = -14.

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