Multiple choice

If $\alpha$ and $\beta$ the roots of the equation $x^2 - q(1 + x) - r = 0$ , then what is $ (1+ \alpha) (1+\beta)$ equal to ?

  1. 1 - r

  2. q - r

  3. 1 + r

  4. q + r
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A Correct answer
Explanation

The equation is x^2 - qx - q - r = 0. The product of roots (1 + alpha)(1 + beta) = 1 + alpha + beta + alpha*beta. From the equation, alpha + beta = q and alpha*beta = -(q + r). Substituting these: 1 + q - q - r = 1 - r.

AI explanation

Rearranging the equation yields x^2 - qx - (q + r) = 0. By Vieta's formulas, the sum of the roots alpha + beta equals q, and the product alpha beta equals -(q + r). Expanding (1 + alpha)(1 + beta) gives 1 + (alpha + beta) + alpha beta, which substitutes as 1 + q - q - r. The result is 1 - r.

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