Question

For \((k \neq 0)\), if the quadratic equation \(kx(x - 2) + 6 = 0\) has two equal roots, then find the value of \(k\).

• A. 2
• B. 4
• C. 6
• D. 8

Hint:

For a quadratic equation to have equal roots, the discriminant \(\Delta = b^2 - 4ac\) must be zero.

Answer 1

The Correct Option is C

First, expand the given equation to write it in the standard quadratic form $ax^2 + bx + c = 0$.
$kx(x-2) + 6 = 0$
$kx^2 - 2kx + 6 = 0$.
Here, the coefficients are $a = k$, $b = -2k$, and $c = 6$.
For a quadratic equation to have two equal roots, its discriminant (D) must be equal to zero.
The formula for the discriminant is $D = b^2 - 4ac$.
Set the discriminant to zero: $b^2 - 4ac = 0$.
Substitute the values of a, b, and c:
$(-2k)^2 - 4(k)(6) = 0$.
$4k^2 - 24k = 0$.
Factor out the common term $4k$:
$4k(k - 6) = 0$.
This gives two possible solutions for k: $4k = 0$ or $k - 6 = 0$.
So, $k = 0$ or $k = 6$.
The problem states that $k \neq 0$.
Therefore, the only valid value for k is 6.