Question
Wed February 23, 2011

# Solve :

Wed February 23, 2011
Dear Student,
Let us take the general form of a quadratic equation:

ax2 + bx + c = 0     [a Ã¢Â‰Â  0 ]                      (1)

If and ÃƒÂŸ be the roots of the equation (1), then

= and ÃƒÂŸ =

Now, suppose that a, b and c are real and rational. Then, the nature of the roots and ÃƒÂŸ of
equation (1) is determinedby the expression (b2 Ã¢Â€Â“ 4ac) under the radical sign.

Therefore (b2 Ã¢Â€Â“ 4ac) is known as the Discriminant of equation (1). Referred to this discriminant following conclusions can be drawn about the nature of roots and ÃƒÂŸ of equation (1):

1. If discriminant is positive (that is, if b2 Ã¢Â€Â“ 4ac > 0), then the roots and ÃƒÂŸ of equation (1) are real andunequal.

2. If discriminant is zero (that is, if b2 Ã¢Â€Â“ 4ac = 0), then the roots and ÃƒÂŸ of equation (1) are real and equal
3. If discriminant is negative (that is, if b2 Ã¢Â€Â“ 4ac < 0), then the roots and ÃƒÂŸ of equation (1) are imaginary and unequal
4. If discriminant is positive and a perfect square then the roots of equation (1) are real, rational and unequal. And if the discriminant is positive but not a perfect square then the roots of equation (1) are real, irrational and unequal.
5. If b2 Ã¢Â€Â“ 4ac is a perfect square but any one of a or b is irrational then the roots of equation (1) are irrational.

Now, here if c is < 0 it means it is a negative digit then discriminant (b2-4ac) becomes positive. Hence the nature of roots will either follow point (1) or point(4) mentioned above.

Regards

Team Topperlearning

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