Find the value of k for which one root of the quadratic equation kx2 14x 8 = 0 is 6 times the other

k = 3.

Step-by-step explanation:

If the 2 roots are A and B we have the relations:

AB = 8/k and A+B = - 14/k.

We are given that A = 6B so

6B^2 = 8/k

B^2 = 8/6k = 4/3k

B = 2 / √ (3k) ... (1)

Now A + B = - 14/k so

6B + B = 7B = - 14/k

B = - 2/k ... (2)

Eliminating B from equations (1) and (2):

2 / √ (3k) = - 2/k

Cross multiply:

2k = - 2√ (3k)

Squaring both sides:

4k^2 = 4 * 3k

4k^2 = 12k

k^2 = 3k

k = 3.