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Roots of a Quadratic Equation and their Nature

Given is the quadratic equation ax2 + bx + c = 0, where a ≠ 0.
 
Assuming that a and b are the roots of this equation.
Then Description: 2122.png 
It can be seen that the nature of both these two roots depends upon the value of b2 – 4ac.
(b2 – 4ac) is also known as Discriminant (D) of the quadratic equation ax2 + bx + c = 0.
If D = 0, thenDescription: 2131.png
So, the roots will be real and equal.
 
If D > 0, thenDescription: 2140.png 
So, the roots will be real and distinct.
 
If D < 0, thenDescription: 2149.png is not real.
 
So, the roots will not be real.
If D is a perfect square (including D = 0) and ab and c are rational, then the roots will also be rational.
 
Example
Given is the quadratic equation ax2 + bx + 1 = 0, where ab (1, 2, 3, 4). For how many set of values of (ab), the quadratic equation ax2 + bx + 1 = 0 will have real roots? (CAT 2003)
Solution
For the roots to be real, D ≥ 0.
D = b2 – 4a ≥ 0.
Forming the table for the above-written condition:
 
b a
1 Not possible
2 1
3 1
3 2
4 1
4 2
4 3
4 4
 
So, there are seven set of values for (ab).
 





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