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

Step 1: General form of the quadratic equation.
The given quadratic equation is: \[ kx(x - 2) + 6 = 0. \] Expanding the equation: \[ kx^2 - 2kx + 6 = 0. \]

Step 2: Condition for equal roots.
For a quadratic equation \(ax^2 + bx + c = 0\) to have equal roots, the discriminant \(\Delta\) must be zero. The discriminant is given by: \[ \Delta = b^2 - 4ac. \] Here, \(a = k\), \(b = -2k\), and \(c = 6\). Substituting these values into the discriminant formula: \[ \Delta = (-2k)^2 - 4(k)(6) = 4k^2 - 24k. \] For the roots to be equal, \(\Delta = 0\), so: \[ 4k^2 - 24k = 0. \]

Step 3: Solve for \(k\).
Factorizing the equation: \[ 4k(k - 6) = 0. \] Thus, \(k = 0\) or \(k = 6\). Since \(k \neq 0\), we conclude that: \[ k = 6. \]

Final Answer: 6.