Multiple choice

Find the greatest value of a non-negative real number $\lambda$ for which both the equations $2x^{2} + (\lambda - 1)x + 8 = 0$ and $x^{2} - 8x + \lambda + 4 = 0$ have real roots.

  1. $\lambda=12$
  2. $\lambda=16$
  3. $\lambda=15$
  4. $\lambda=14$
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A Correct answer
Explanation

For real roots, the discriminant D >= 0. For the first equation: (lambda - 1)^2 - 4(2)(8) >= 0 => (lambda - 1)^2 >= 64 => lambda - 1 >= 8 or lambda - 1 <= -8. So lambda >= 9 or lambda <= -7. For the second: (-8)^2 - 4(1)(lambda + 4) >= 0 => 64 - 4lambda - 16 >= 0 => 48 >= 4lambda => lambda <= 12. Combining these, the range is 9 <= lambda <= 12. The greatest value is 12.

AI explanation

For the first equation 2x^2 + (λ - 1)x + 8 = 0 to have real roots, the discriminant must be non-negative, so (λ - 1)^2 - 64 >= 0, which gives λ - 1 >= 8 or λ - 1 <= -8, meaning λ >= 9 or λ <= -7. Since λ is non-negative, λ >= 9. For the second equation x^2 - 8x + λ + 4 = 0, the discriminant condition is 64 - 4(1)(λ + 4) >= 0, which simplifies to 64 - 4λ - 16 >= 0, giving λ <= 12. Combining λ >= 9 and λ <= 12 gives 9 <= λ <= 12, so the greatest value is λ = 12.

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