Multiple choice

The positive value of $k$ for which the equations $x^2 + kx + 64 = 0$ and $x^2- 8x + k = 0$ will both have real roots is ________.

  1. $4$
  2. $8$
  3. $12$
  4. $16$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

For real roots, D >= 0. For x^2 + kx + 64 = 0, k^2 - 256 >= 0, so |k| >= 16. For x^2 - 8x + k = 0, 64 - 4k >= 0, so k <= 16. Both conditions are satisfied only when k = 16.

AI explanation

For real roots, the discriminant b^2 - 4ac must be non-negative. For x^2 + kx + 64 = 0, this means k^2 - 256 >= 0, so k^2 >= 256; for x^2 - 8x + k = 0, this means 64 - 4k >= 0, so k <= 16. Testing the positive options shows that k = 16 satisfies both conditions because 16^2 >= 256 and 16 <= 16. Therefore, the positive value is 16.

Keep practicing — related questions