Multiple choice

If $- 4$ is a root of the equation $x ^ { 2 } + p x - 4 = 0$ and the equation $x ^ { 2 } + p x + q = 0$ has equal roots, then the values of $p$ and $q$ are, respectively

  1. $- 3$ and $\dfrac { 9 } { 4 }$
  2. $3$ and $\dfrac { 9 } { 4 }$
  3. $\dfrac { 9 } { 4 }$ and $3$
  4. $4$ and $3$
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B Correct answer
Explanation

Since -4 is a root of x^2 + px - 4 = 0, (-4)^2 + p(-4) - 4 = 0, so 16 - 4p - 4 = 0, 4p = 12, p = 3. For x^2 + px + q = 0 to have equal roots, the discriminant D = p^2 - 4q = 0. With p = 3, 9 - 4q = 0, so q = 9/4.

AI explanation

Substituting the root -4 into the first equation x^2 + px - 4 = 0 gives 16 - 4p - 4 = 0. Solving this gives 12 = 4p, so p = 3. For the second equation x^2 + px + q = 0 to have equal roots, its discriminant must be zero, meaning p^2 - 4q = 0. Substituting p = 3 gives 9 - 4q = 0, which means q = 9/4.

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