Question

9. Given one of the root of the quadratic equation x^2 - 5kx+ k=0 is four times the other root, find the value of k

Answer 1

k = 1/4.

Explanation:

Let the roots be r and 4r and using the fact that sum of the roots = -b/a and product = a/c, we have:

r + 4r = -(-5k)

r*4r = k

5r = 5k

4r^2 = k

From the 3rd equation:

r = k, therefore:

4k^2 = k

4k = 1

k = 1/4.

Answer 2

k=1/4

Explanation:

x² - 5kx+ k=0 ....(1)

suppose 2 roots: a and 4a

(x-a)(x-4a)=0

x² - 5ax + 4a² = 0 ....(2)

compare (1) and (2): -5a = -5k a=k

4a² = k 4a² = a (substitute)

a(4a-1) = 0

a = 0 or a = 1/4 (if a=0 k=0 and x=0 is not the solution)

a = 1/4 k = 1/4

(a₂ = 1/4 * 4 = 1)