Multiple choice

If both the roots of the equation x2 - 6x + k = 0 are positive integers, how many values will k take?

  1. 1

  2. 2

  3. 3

  4. 4

  5. Infinite
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

x^2 - 6x + k = 0. Roots are integers. Sum of roots = 6, product = k. Possible pairs (x1, x2) such that x1+x2=6: (1,5) -> k=5; (2,4) -> k=8; (3,3) -> k=9. There are 3 possible values for k.

AI explanation

Let the positive integer roots of the equation x^2 - 6x + k = 0 be p and q. From Vieta's formulas, the sum of the roots is p + q = 6. The only pairs of positive integers that add up to 6 are (1, 5), (2, 4) and (3, 3). Therefore, the product of the roots k, which equals p times q, can take the three distinct values of 5, 8 and 9.

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