Multiple choice

If the equations $2x^{2}+x+k=0$ and $x^{2}+\dfrac{x}{2}-1=0$ have a common root, then the value of $k$ is

  1. $1$
  2. $3$
  3. $-1$
  4. $-2$
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D Correct answer
Explanation

If the equations have a common root x, then 2x^2 + x + k = 0 and x^2 + 0.5x - 1 = 0. Multiplying the second equation by 2 gives 2x^2 + x - 2 = 0. Comparing this to 2x^2 + x + k = 0, we see that k must be -2.

AI explanation

Multiply the second equation, x^2 + 0.5x - 1 = 0, by 2 to get 2x^2 + x - 2 = 0. Comparing this directly with the first equation, 2x^2 + x + k = 0, we can see they share the exact same coefficients for x^2 and x. For these two equations to have a common root without being identical, their constant terms must also match exactly. Therefore, k must equal -2.

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