The condition that one root of the equation $a{x^2} + bx + c$ exceeds the other by p is:
- ${a^2}{p^2} = 4ac\,$
- ${a^2}{p^2} = {b^2}4ac$
- ${a^2}{p^2} = {b^2} - 4ac$
- $ap = {b^2} + 4ac$
If roots are alpha and alpha + p, then their sum is -b/a and product is c/a. (alpha + p) - alpha = p. The difference of roots is sqrt(D)/|a| = p. Thus, D/a^2 = p^2, which means (b^2 - 4ac) / a^2 = p^2, or b^2 - 4ac = a^2 * p^2.
Let the roots of the quadratic equation ax^2 + bx + c = 0 be alpha and beta. The sum of the roots is alpha + beta = -b/a and the product is alpha * beta = c/a. The difference between the roots is given by |alpha - beta| = sqrt((alpha + beta)^2 - 4*alpha*beta). Substituting the sum and product into the difference formula gives p = sqrt((-b/a)^2 - 4(c/a)). Squaring both sides results in p^2 = (b^2/a^2) - (4c/a), which simplifies to p^2 = (b^2 - 4ac)/a^2. By cross-multiplying, the required condition is a^2*p^2 = b^2 - 4ac.