Multiple choice

The equation $\displaystyle x^{2}-6x+8+\lambda (x^{2}-4x+3)=0,\lambda \in R$, has

  1. real and unequal roots for all $\lambda $
  2. real roots for $\lambda < 0$ only
  3. real roots for $\lambda > 0$ only
  4. real and unequl roots for $\lambda =0$ only
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A Correct answer
Explanation

The discriminant of the quadratic equation (1+lambda)x^2 - (6+4lambda)x + (8+3lambda) = 0 is D = (6+4lambda)^2 - 4(1+lambda)(8+3lambda). Simplifying this yields D = 16lambda^2 + 48lambda + 36 - 4(3lambda^2 + 11lambda + 8) = 4lambda^2 + 4lambda + 4. Since the discriminant 4(lambda^2 + lambda + 1) is always positive for all real lambda, the roots are always real and unequal.

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