Question

For what value of k, the roots of the quadratic equation $kx(x - 2\sqrt 5 ) + 10 = 0$ are equal?

Answer 1

Hint: Find the discriminant value and check when it is equal to 0 i.e. D=0.

Complete step-by-step answer:
We have
 $
  kx(x - 2\sqrt 5 ) + 10 = 0 \to 1 \\
  k{x^2} - 2\sqrt 5 kx + 10 = 0 \\
  {\text{Given roots are equal then}} \\
  D = {b^2} - 4ac = 0 \Rightarrow {b^2} = 4ac \\
   \Rightarrow {(2\sqrt 5 k)^2} = 4 \times 10 \times k \\
   \Rightarrow 20{k^2} = 40k \\
   \Rightarrow 20k(k - 2) = 0 \\
  {\text{But k}} \ne 0,{\text{ }}\therefore k = 2 \\
  \therefore {\text{ for k = 2 roots of given quadratic equation are equal}}{\text{.}} \\ $

Note: Whether the roots are for real & different, real & equal or imaginary it depends on the value of the discriminant. In this problem we are concerned with discriminant value 0 (which fetches real & equal roots) to get the value of k.