Question

If the values of \( k \) for which the equation \( x^2 + 2(k+2)x + 6k + 7 = 0 \) has equal roots are \( k_1 \) and \( k_2 \), then \( k_1^2 + k_2^2 \) is:

• A. \( 8 \)
• B. \( 9 \)
• C. \( 10 \)
• D. \( 12 \)

Hint:

For equal roots, set the discriminant equal to zero and solve for \( k \).

Answer 1

The Correct Option is C

For equal roots, set the discriminant to zero: \[ \Delta = \left[ 2(k+2) \right]^2 - 4 \times 1 \times (6k + 7) = 0 \] Simplifying the discriminant: \[ \Delta = 4(k+2)^2 - 4(6k + 7) = 4k^2 - 8k - 12 \] Setting the discriminant equal to zero: \[ 4k^2 - 8k - 12 = 0 \] Dividing by 4: \[ k^2 - 2k - 3 = 0 \] Solving the quadratic: \[ (k - 3)(k + 1) = 0 \quad \Rightarrow \quad k_1 = 3, \quad k_2 = -1 \] Now calculate \( k_1^2 + k_2^2 \): \[ k_1^2 + k_2^2 = 9 + 1 = 10 \]