Posted by: catindiaonline | October 25, 2008

CAT: Quadratic Expressions & Equations

QUADRATIC EXPRESSION: If a \not=0, b, c are complex numbers then  ax^2 + bx +c is called a quadratic expression in x.


QUADRATIC EQUATION: If a \not=0,b,c are complex numbers then  ax^2 + bx + c = 0 is called a quadratic equation in x.


ROOT OF A QUADRATIC EQUATION: If a\alpha^2 + b\alpha + c = 0then \alpha is a root or solution of the quadratic equation ax^2 + bx + c = 0.

A quadratic equation can not have more than two roots or two solutions. The roots of ax^2 + bx + c = 0 are \frac{-b\pm \sqrt{b^2 - 4ac}}{2a} and its discrminent is \triangle = b^2 -4ac.


NATURE OF THE ROOTS OF THE EQUATION ax^2 + bx + c = 0

1.If a,b,c are real and \triangle>0, then the roots are real and distinct.

2.If a,b,c are real and \triangle=0, then the roots are real and equal.

3.If a,b,c are real and \triangle<0, then the roots are two conjugate complex numbers.

4.If a,b,c are rational and \triangle>0, and is a perfect square then the roots are rational and distinct.

5.If a,b,c are rational and \triangle>0, and is not a perfect square then the roots are conjugate surds i.e \alpha\pm \beta.

6.If a,b,c are rational and \triangle<0, then the roots are conjugate complex numbers i.e, \alpha\pm i\beta.


FORMATION OF THE QUADRATIC EQUATION WITH ROOTS \alpha AND \beta: The quadratic equation whose roots are \alpha and \beta is x^2 - (\alpha + \beta)x + \alpha \beta = 0 \Rightarrow (x-\alpha)(x-\beta) = 0.


RELATION BETWEEN THE ROOTS \alpha , \beta OF ax^2 + bx + c = 0.

1. \alpha + \beta = \frac{-b}{a} ,   \alpha \beta = \frac{c}{a}.

2.|\alpha - \beta| = \frac {\sqrt{b ^2 - 4ac}}{|a|}.

3. \alpha ^2 + \beta ^2 = \frac{b ^2-2ac}{a ^2}.

4. \alpha ^3 + \beta ^3 = \frac{3abc-b ^3}{a ^3}.

5. \frac{1}{\alpha} + \frac{1}{\beta} = \frac{-b}{c}.

6. \frac{1}{\alpha ^2} + \frac{1}{\beta ^2} = \frac{b ^2 -2ac}{c ^2}.

7. \frac{1}{(a\alpha + b)} + \frac{1}{(a\beta+b)} = \frac{b}{ac}.

8. \frac{1}{(a\alpha + b)  ^2} + \frac{1}{(a\beta +b) ^2} = \frac{ b ^2 -2ac}{a ^2 c ^2}.

9. \frac{1}{(a\alpha + b)  ^3} + \frac{1}{(a\beta +b) ^3} = \frac{ b ^3 -3abc}{a ^3 c ^3}.

10. |\alpha ^2 - \beta ^2| = \frac{|b|\sqrt{b ^2 -4ac}}{a ^2}.

For rest go to CAT India Online’s CATclub

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