Multiple choice

If $-5$ is a root of the quadratic $2{x^2} + px - 15 = 0$ and the quadratic equation $p({x^2} + x) + k = 0\,\,$ has equal root , find the value of $k.$

  1. $\dfrac {7}{4}$
  2. $\dfrac {7}{2}$
  3. $-\dfrac {7}{4}$

  4. None of these
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A Correct answer
Explanation

If -5 is a root of 2x^2 + px - 15 = 0, then 2(25) - 5p - 15 = 0, so 50 - 15 = 5p, 35 = 5p, p = 7. The second equation is 7(x^2 + x) + k = 0, or 7x^2 + 7x + k = 0. For equal roots, discriminant D = b^2 - 4ac = 0. So 7^2 - 4(7)(k) = 0, 49 - 28k = 0, k = 49/28 = 7/4.

AI explanation

Because -5 is a root of 2x^2 + px - 15 = 0, substitute x = -5 to get 2(25) - 5p - 15 = 0. Solving this gives 50 - 5p - 15 = 0, so p = 7. Substitute p = 7 into the second equation to get 7(x^2 + x) + k = 0, which rearranges to 7x^2 + 7x + k = 0. For this equation to have equal roots, its discriminant must be zero, so b^2 - 4ac = 49 - 28k = 0. Solving for k gives k = 49/28, which simplifies to 7/4. The result is 7/4.

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