The value of k for which the quadratic equation kx(x−2)+6=0 has equal roots is –

To find the value of k for which the quadratic equation kx(x - 2) + 6 = 0 has equal roots, we follow these steps based on algebraic principles.
Step-by-Step Analysis
Standard Form Conversion:

Expand the equation: kx(x - 2) + 6 = 0 becomes kx^2 - 2kx + 6 = 0.

Apply Discriminant Condition:

A quadratic equation in the form ax^2 + bx + c = 0 has equal roots if its discriminant (D = b^2 - 4ac) equals zero.
Here, a = k, b = -2k, and c = 6.

Substitute these into the discriminant formula: D = (-2k)^2 - 4(k)(6) = 0.

Solve for k:

4k^2 - 24k = 0.

Factor the expression: 4k(k - 6) = 0.

This gives two possible values for k: 4k = 0 (so k = 0) or k - 6 = 0 (so k = 6).

If k = 0, the equation is no longer quadratic (it becomes 6 = 0), so we must choose k = 6.