Multiple choice

If $a,\beta \in R$ and the quadratic equation ${x^2} + 2x + 7 = 0$ and $4{x^2} + ax + \beta = 0$ have a common root,then the value of $a + \beta $ is

  1. -$4$
  2. $36$
  3. $20$
  4. $24$
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B Correct answer
Explanation

Let the common root be k. Then k^2 + 2k + 7 = 0 and 4k^2 + ak + beta = 0. Multiplying the first by 4 gives 4k^2 + 8k + 28 = 0. Comparing this to the second equation, we get a = 8 and beta = 28. Thus, a + beta = 36.

AI explanation

The discriminant of the first quadratic equation x^2 + 2x + 7 = 0 is 2^2 - 4(1)(7), which equals -24. Since the discriminant is negative, the roots are complex, and for two quadratic equations to have a common complex root, they must share the exact same set of roots. Let the roots be alpha and beta, and by Vieta's formulas for the first equation, alpha + beta = -2 and alpha times beta = 7. For the second equation 4x^2 + ax + beta = 0, the sum of the roots is -a divided by 4 and the product is beta divided by 4. Equating the sums gives -a divided by 4 = -2, so a = 8. Equating the products gives beta divided by 4 = 7, so beta = 28. Adding these values together, the sum a + beta equals 8 + 28, which is 36.

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