Multiple choice

If quadratic equations ${ 2x }^{ 2 }-3x+5=0$ and ${ ax }^{ 2 }-bx+c=0,a,b,c\epsilon N$ have a common root then the value of a+b+c can be equal to

  1. 10

  2. 15

  3. 20

  4. 25
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A Correct answer
Explanation

For a common root, the equations must be proportional if the roots are complex or satisfy the condition for a shared root. Given a, b, c are natural numbers, testing the ratio 2/a = -3/-b = 5/c. If a=2, b=3, c=5, then a+b+c = 10.

AI explanation

The discriminant of 2x^2 - 3x + 5 = 0 is (-3)^2 - 4(2)(5) = 9 - 40 = -31, which is less than zero, indicating the roots are strictly non-real complex numbers. For ax^2 - bx + c = 0 to share a root, it must be a scalar multiple of the first equation. Let ax^2 - bx + c = k(2x^2 - 3x + 5) = 2kx^2 - 3kx + 5k. Thus, a = 2k, b = 3k, and c = 5k. The sum a + b + c = 2k + 3k + 5k = 10k. Since a, b, and c must be natural numbers, k must be a natural number. For k = 1, the sum is 10.

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