Multiple choice

The equations $\displaystyle 4x^{2}-x-1= 0$ and $\displaystyle 3x^{2}+\left ( \lambda +\mu \right )x+\lambda-\mu = 0$ have at least one common root , where $\lambda$ and $\mu$ are rational numbers. Find the value of $\lambda$ and $\mu$.

  1. $\displaystyle \lambda = 0, \mu =-\frac{3}{4}$
  2. $\displaystyle \lambda = -\frac{3}{4}, \mu = 0$
  3. $\displaystyle \lambda = -\frac{3}{4}, \mu = \frac{3}{4}$
  4. $\displaystyle \lambda = -\frac{3}{4}, \mu = \frac{1}{4}$
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B Correct answer
Explanation

The roots of 4x^2 - x - 1 = 0 are x = (1 +/- sqrt(1 + 16))/8 = (1 +/- sqrt(17))/8. For the second equation to share a root, substituting x = (1 + sqrt(17))/8 into 3x^2 + (lambda + mu)x + (lambda - mu) = 0 must hold. Since lambda and mu are rational, the irrational parts must cancel, leading to lambda = -3/4 and mu = 0.

AI explanation

Let the common root of the two equations be r. From the first equation, 4r^2 - r - 1 = 0, which means r is irrational. Since lambda and mu are given as rational numbers, the second equation 3x^2 + (lambda + mu)x + lambda - mu = 0 must also have its second root equal to r, making both roots identical to the roots of the first equation. Thus, the ratios of the corresponding coefficients of the two equations must be equal: 3/4 = (lambda + mu)/(-1) = (lambda - mu)/(-1). From (lambda + mu)/(-1) = 3/4, we get lambda + mu = -3/4. From (lambda - mu)/(-1) = 3/4, we get lambda - mu = -3/4. Adding these two equations gives 2(lambda) = -3/2, so lambda = -3/4. Substituting this into the first equation yields mu = 0.

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