Multiple choice

Number of non-negative integral values of $'k'$ for which roots of the equation $x^2+6x+k=0$ are rational is-

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

Roots of x^2+6x+k=0 are rational if the discriminant D = 6^2 - 4k = 36 - 4k is a perfect square. For k >= 0: k=0 (D=36, yes), k=2 (D=28, no), k=5 (D=16, yes), k=8 (D=4, yes), k=9 (D=0, yes). The values are 0, 5, 8, 9. There are 4 such values.

AI explanation

Using the discriminant condition for rational roots, we set the discriminant of x squared plus 6x plus k equals 0 to a perfect square. This gives 36 minus 4k equals p squared, or 9 minus k equals the square of p divided by 4. For non-negative k, testing integer values for k reveals that k equals 9 gives p equals 0, k equals 8 gives p equals plus or minus 2, k equals 5 gives p equals plus or minus 4, and k equals 0 gives p equals plus or minus 6. All four values of k yield perfect square discriminants, resulting in rational roots.

Keep practicing — related questions