Question

If the roots of the equation \(x^2 - 2(1+3k)x + 7(3+2k) = 0\) are equal, where \(k<0\), then which of the following is true?

• A. \(9k^2 - k + 2 = 0\)
• B. \(k^2 - 4 = 0\)
• C. \(k^2 + k - 8 = 0\)
• D. \(9k^2 + k - 10 = 0\)

Hint:

For quadratic equations with equal roots, always check the discriminant (\(b^2 - 4ac\)) and set it equal to zero.

Answer 1

The Correct Option is D

The condition for the roots of a quadratic equation to be equal is that its discriminant must be zero. The general form of a quadratic equation is \(ax^2 + bx + c = 0\), and the discriminant is given by: \[ \Delta = b^2 - 4ac \] For the given quadratic equation, we have: - \(a = 1\), - \(b = -2(1 + 3k)\), - \(c = 7(3 + 2k)\). The discriminant will be: \[ \Delta = \left(-2(1 + 3k)\right)^2 - 4 \cdot 1 \cdot 7(3 + 2k) \] Simplifying this expression: \[ \Delta = 4(1 + 3k)^2 - 28(3 + 2k) \] Solving this, we get the quadratic equation: \[ 9k^2 + k - 10 = 0 \] Thus, the correct answer is \(9k^2 + k - 10 = 0\).