Multiple choice

If the roots of the equation $x^2+ px + q = 0$ are $r_1$ and $r_2$, then $r_1^2 + r_2^2=$

  1. $p^2+q^2$
  2. $p^2-2q$
  3. $p^2+2q^2$
  4. $p^2$
  5. $q^2$
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B Correct answer
Explanation

r1+r2 = -p, r1*r2 = q. r1^2 + r2^2 = (r1+r2)^2 - 2r1r2 = (-p)^2 - 2q = p^2 - 2q.

AI explanation

Using Vieta's formulas for the equation x^2 + px + q = 0, the sum of the roots is r1 + r2 = -p and the product of the roots is r1 * r2 = q. The expression r1^2 + r2^2 can be rewritten using the algebraic identity r1^2 + r2^2 = (r1 + r2)^2 - 2r1r2. Substituting the known sum and product gives (-p)^2 - 2(q), which simplifies to p^2 - 2q.

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