If sum of the roots is p and q and the sum of their squares is $ q^{2} $ , the equation is
- $ \displaystyle x^{2}-px+p^{2}q^{3}=0 $
- $ \displaystyle x^{2}-px+q^{3}=0 $
- $ \displaystyle x^{2}-px+\frac{p^{2}-q^{2}}{2}=0 $
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None of these
Let roots be a, b. a+b = p. a^2+b^2 = q^2. We know (a+b)^2 = a^2 + b^2 + 2ab. p^2 = q^2 + 2ab. ab = (p^2 - q^2) / 2. The quadratic equation is x^2 - (sum)x + (product) = 0, which is x^2 - px + (p^2 - q^2) / 2 = 0.
Let the roots be alpha and beta, with their sum given as p, so alpha plus beta equals p. The sum of their squares is given as q squared, and using the identity for the sum of squares, we have alpha squared plus beta squared equals (alpha plus beta) squared minus 2 times alpha times beta. Substituting the known sum, q squared equals p squared minus 2 times alpha beta, which means the product alpha beta equals (p squared - q squared) divided by 2. The required quadratic equation is x squared minus the sum of the roots times x plus the product of the roots equals 0, which is x squared - px + (p squared - q squared)/2 = 0.