Question

Write all the values of k for which the quadratic equation x² + kx + 16 = 0 has equal roots. Find the roots of the equation so obtained.

Answer 1

Given: x² + kx + 16 = 0

Step-wise calculation:

1. For equal (repeated) roots the discriminant D = b² – 4ac must be 0.

2. Here a = 1, b = k, c = 16.

So D = k² – 4(1)(16) = k² – 64.

3. Set D = 0: k² – 64 = 0

⇒ k² = 64

⇒ k = 8 or k = –8

4. The repeated root for a quadratic ax² + bx + c = 0 is `x = -b/(2a)`.

So `x = −k/(2 xx 1) = -k/2`.

If k = 8, root = `-8/2` = –4 (double root).

If k = –8, root = `-(-8)/2` = 4 (double root).

Values of k: k = 8 and k = –8. 

Corresponding repeated roots: for k = 8 → x = −4 (double); for k = –8 → x = 4 (double).