Multiple choice

If $\alpha ,\beta $ are the roots of the equation $ax^{2}+bx+c=0$, then the value of $\displaystyle \frac{1}{a\alpha +b}+\frac{1}{a\beta +b}$ is

  1. $\displaystyle \frac{a}{bc}$
  2. $\displaystyle \frac{{b} }{{a} c}$
  3. $\displaystyle \frac{c}{ab}$
  4. $\displaystyle \frac{ab}{c}$
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B Correct answer
Explanation

Given roots alpha, beta, alpha+beta = -b/a and alpha*beta = c/a. The expression is (a*beta+b + a*alpha+b) / ((a*alpha+b)(a*beta+b)). Numerator: a(alpha+beta) + 2b = a(-b/a) + 2b = -b + 2b = b. Denominator: a^2(alpha*beta) + ab(alpha+beta) + b^2 = a^2(c/a) + ab(-b/a) + b^2 = ac - b^2 + b^2 = ac. Result is b/(ac).

AI explanation

For the quadratic equation ax^2 + bx + c = 0, the sum of the roots is alpha + beta = -b/a and the product is alpha*beta = c/a. Combining the fractions over a common denominator gives (a*alpha + a*beta + 2b) / ((a*alpha + b)(a*beta + b)). Expanding the denominator yields a^2*alpha*beta + ab*alpha + ab*beta + b^2, which factors to a^2*alpha*beta + ab(alpha + beta) + b^2. Substituting the sum and product values gives a^2(c/a) + ab(-b/a) + b^2 = ac - b^2 + b^2 = ac. The numerator simplifies to a(-b/a) + 2b = b. Therefore, the entire expression evaluates to b / (ac).

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