Multiple choice

Let us consider a quadratic equation $x^2+\lambda x+\lambda +1.25=0$, where $\lambda$ is a constant. The value of $\lambda$ such that the above quadratic equation has two coincident roots

  1. $\lambda =5$ or $\lambda=-1$
  2. $\lambda =1$ or $\lambda=5$
  3. $\lambda =-5$ or $\lambda=1$

  4. None of these
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A Correct answer
Explanation

For a quadratic equation to have coincident roots, the discriminant (D = b^2 - 4ac) must be zero. For x^2 + lambda*x + (lambda + 1.25) = 0, D = lambda^2 - 4(lambda + 1.25) = lambda^2 - 4*lambda - 5 = 0. Solving (lambda - 5)(lambda + 1) = 0 gives lambda = 5 or lambda = -1.

AI explanation

For a quadratic equation to have two coincident roots, its discriminant must equal zero, so we set D = b^2 - 4ac = 0. For x^2 + (lambda)x + (lambda + 1.25) = 0, this gives the equation lambda^2 - 4(lambda + 1.25) = 0. Expanding and simplifying leads to lambda^2 - 4*lambda - 5 = 0. Factoring this quadratic gives (lambda - 5)(lambda + 1) = 0, meaning lambda = 5 or lambda = -1.

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