Question:

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

Updated On: Apr 8, 2024
  • p2−4q {{p}^{2}}-4q
  • (p2−4q)2 {{({{p}^{2}}-4q)}^{2}}
  • p2+4q {{p}^{2}}+4q
  • (p2+4q)2 {{({{p}^{2}}+4q)}^{2}}
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The Correct Option is A

Solution and Explanation

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

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Concepts Used:

Complex Numbers and Quadratic Equations

Complex Number: Any number that is formed as a+ib is called a complex number. For example: 9+3i,7+8i are complex numbers. Here i = -1. With this we can say that i² = 1. So, for every equation which does not have a real solution we can use i = -1.

Quadratic equation: A polynomial that has two roots or is of the degree 2 is called a quadratic equation. The general form of a quadratic equation is y=ax²+bx+c. Here a≠0, b and c are the real numbers.