Multiple choice

lf $\alpha,\ \beta$ are the roots of $x^{2}-px +q=0$, then the equation whose roots are $\alpha\beta+\alpha+\beta,\ \alpha\beta-\alpha-\beta$, is:

  1. $ x^{2}+2qx+p^{2}+q^{2}=0$
  2. $ x^{2}-2qx +p^{2}+q^{2}=0$
  3. $ x^{2}-2qx +q^{2}-p^{2}=0$
  4. $ x^{2}+2qx +q^{2}-p^{2}=0$
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C Correct answer
AI explanation

From the given equation x^2 - px + q = 0, the sum of the roots is alpha + beta = p and the product is alpha * beta = q. The new roots are alpha*beta + alpha + beta = q + p and alpha*beta - alpha - beta = q - p. The sum of the new roots is (q + p) + (q - p) = 2q, and the product of the new roots is (q + p)(q - p) = q^2 - p^2. Forming the new quadratic equation x^2 - (sum)x + (product) = 0 gives x^2 - 2qx + q^2 - p^2 = 0.

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