Question

If one root of the equation \( x^2 + px + 12 = 0 \) is \( 4 \), while the equation \( x^2 + px + q = 0 \) has equal roots, then the value of \( q \) is:

• A. \( 4 \)
• B. \( 12 \)
• C. \( 3 \)
• D. \( \frac{49}{4} \)

Hint:

For quadratic equations: - If a root is known, substitute it into the equation to find unknown coefficients. - For equal roots, use the condition \( \Delta = 0 \), where \( \Delta = b^2 - 4ac \).

Answer 1

The Correct Option is D

Step 1: Find the value of \( p \). The given quadratic equation is: \[ x^2 + px + 12 = 0. \] Since one root is given as \( x = 4 \), substituting it into the equation: \[ 4^2 + 4p + 12 = 0. \] \[ 16 + 4p + 12 = 0. \] \[ 4p + 28 = 0. \] \[ 4p = -28. \] \[ p = -7. \] Step 2: Use the condition for equal roots in the second equation. The second equation given is: \[ x^2 + px + q = 0. \] For equal roots, the discriminant must be zero: \[ \Delta = p^2 - 4q = 0. \] Substituting \( p = -7 \): \[ (-7)^2 - 4q = 0. \] \[ 49 - 4q = 0. \] \[ 4q = 49. \] \[ q = \frac{49}{4}. \] Thus, the correct answer is \( \frac{49}{4} \).