Question

{"given":"One root of the quadratic equation $x^2 + px + 12 = 0$ is 4. The second equation $x^2 + px + q = 0$ has equal roots (discriminant = 0). ","key_observation":"When a value is a root of a quadratic equation, substituting it satisfies the equation. For equal roots, the discriminant $\\Delta = b^2 - 4ac = 0$. We can find $p$ from the first condition and use it in the discriminant condition for the second equation.","option_analysis":[{"label":"(A)","text":"$\\frac{49}{4}$","verdict":"correct","explanation":"Since 4 is a root of $x^2 + px + 12 = 0$, we get $16 + 4p + 12 = 0$, so $p = -7$. For equal roots in $x^2 + px + q = 0$, discriminant = 0: $p^2 - 4q = 0$. Substituting $p = -7$: $49 - 4q = 0$, giving $q = \\frac{49}{4}$."},{"label":"(B)","text":"4","verdict":"incorrect","explanation":"This would give discriminant $49 - 16 = 33 \\neq 0$, so the equation would not have equal roots."},{"label":"(C)","text":"3","verdict":"incorrect","explanation":"This would give discriminant $49 - 12 = 37 \\neq 0$, so the equation would not have equal roots."},{"label":"(D)","text":"12","verdict":"incorrect","explanation":"This would give discriminant $49 - 48 = 1 \\neq 0$, so the equation would not have equal roots."}],"answer":"(A)","formula_steps":[]}