Multiple choice

If $\alpha$ ,$\beta$ are the positive roots of a quadratic equation such that $\left | \alpha -\beta \right |=1$ and $\displaystyle \frac{\alpha }{\beta }+\frac{\beta }{\alpha }=\frac{5}{2}$, then the quadratic equation is

  1. $x^{2}+3x+2=0$
  2. $x^{2}-3x+2=0$
  3. $x^{2}-3x-4=0$
  4. $x^{2}-x+2=0$
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B Correct answer
Explanation

Given |alpha - beta| = 1 and alpha/beta + beta/alpha = 5/2. The second equation is (alpha^2 + beta^2) / (alpha*beta) = 5/2. Using (alpha+beta)^2 = alpha^2+beta^2 + 2alpha*beta, we find alpha*beta = 2 and (alpha+beta)^2 = 9, so alpha+beta = 3. The equation is x^2 - 3x + 2 = 0.

AI explanation

Using the given equation alpha/beta + beta/alpha = 5/2, we combine the fractions to get (alpha^2 + beta^2)/(alpha*beta) = 5/2. Applying the algebraic identity alpha^2 + beta^2 = (alpha - beta)^2 + 2*alpha*beta, we substitute the absolute difference alpha - beta = 1 to find the numerator becomes 1 + 2*alpha*beta. This gives the proportion (1 + 2*alpha*beta)/(alpha*beta) = 5/2, which simplifies to 2 + 4*alpha*beta = 5*alpha*beta, meaning the product of the roots is alpha*beta = 2. Using the sum of roots formula, (alpha - beta)^2 = (alpha + beta)^2 - 4*alpha*beta, so 1 = (alpha + beta)^2 - 8, which means (alpha + beta)^2 = 9 and the sum is 3. Forming the quadratic equation x^2 - (sum)x + (product) = 0, we get x^2 - 3x + 2 = 0. The result is x^2 - 3x + 2 = 0.

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