Question

If \( x^2 + 2(K+2)x + 36 = 0 \) has equal roots, then \( K = \)

• A. 1 or -1
• B. 2
• C. 3
• D. 4 or 8

Hint:

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

Answer 1

The Correct Option is D

For the quadratic equation to have equal roots, the discriminant must be zero. The discriminant \( \Delta \) for the equation \( ax^2 + bx + c = 0 \) is given by: \[ \Delta = b^2 - 4ac. \] Here, \( a = 1 \), \( b = 2(K+2) \), and \( c = 36 \). Thus, the discriminant is: \[ \Delta = \left( 2(K+2) \right)^2 - 4 \times 1 \times 36 = 0. \] Simplifying: \[ 4(K+2)^2 - 144 = 0 \quad \Rightarrow \quad 4(K+2)^2 = 144 \quad \Rightarrow \quad (K+2)^2 = 36. \] Solving: \[ K + 2 = 6 \quad \Rightarrow \quad K = 4, \quad \text{or} \quad K + 2 = -6 \quad \Rightarrow \quad K = -8. \] Thus, \( K = 4 \) or \( K = -8 \).