Solveeit Logo
Login

Question

Mathematics Question on Complex Numbers and Quadratic Equations

If x2−px+q=0{{x}^{2}}-px+q=0 has the roots α\alpha and β\beta then the value of (α−β)2{{(\alpha -\beta )}^{2}} is equal to

A

p2−4q{{p}^{2}}-4q

B

(p2−4q)2{{({{p}^{2}}-4q)}^{2}}

C

p2+4q{{p}^{2}}+4q

D

(p2+4q)2{{({{p}^{2}}+4q)}^{2}}

Answer

p2−4q{{p}^{2}}-4q

Explanation

Solution

The correct option is(A): p2−4q.{{p}^{2}}-4q.

α\alpha and β\beta are roots of x2−px+q=0{{x}^{2}}-px+q=0
∵\because α+β=p,αβ=q\alpha +\beta =p,\alpha \beta =q
∴\therefore (α−β)2=(α+β)2−4αβ=p2−4q{{(\alpha -\beta )}^{2}}={{(\alpha +\beta )}^{2}}-4\alpha \beta ={{p}^{2}}-4q