Multiple choice

For m ∈ Z ≥ 0 , the quadratic equation y2 - 8y + m = 0 has two integer roots. How many values of m are possible?

  1. 2

  2. 3

  3. 5

  4. 6
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

For y^2 - 8y + m = 0 to have integer roots, the discriminant D = 64 - 4m must be a perfect square, say k^2. So 4(16-m) = k^2, meaning 16-m must be a perfect square (0, 1, 4, 9, 16). Thus m can be 16, 15, 12, 7, 0. There are 5 possible values.

AI explanation

For the quadratic equation y squared minus 8y plus m equals zero to have two integer roots, the roots must be factors that sum to 8. The possible pairs of roots that add up to 8 are (0, 8), (1, 7), (2, 6), (3, 5), and (4, 4). Since the product of the roots gives the constant term m, these pairs yield the values 0, 7, 12, 15, and 16 respectively. Therefore, there are exactly 5 possible values for m.

Keep practicing — related questions