If $ {{x}^{2}}-px+q=0 $ has the roots $ \alpha $ and $ \beta $ then the value of $ {{(\alpha -\beta )}^{2}} $ is equal to
- $ {{p}^{2}}-4q $
- $ {{({{p}^{2}}-4q)}^{2}} $
- $ {{p}^{2}}+4q $
- $ {{({{p}^{2}}+4q)}^{2}} $
The Correct Option is A
Solution and Explanation
The correct option is(A): \({{p}^{2}}-4q.\)
\(\alpha\) and \(\beta\) are roots of \({{x}^{2}}-px+q=0\)
\(\because\) \(\alpha +\beta =p,\alpha \beta =q\)
\(\therefore\) \({{(\alpha -\beta )}^{2}}={{(\alpha +\beta )}^{2}}-4\alpha \beta ={{p}^{2}}-4q\)
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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.
