


Enter Quadratic Equation 



Discriminant & Nature Of Roots

Rules:
1. If D > 0 then the roots are real and distinct.
(i) If D is a perfect square then
the roots are rational and distinct.
(ii) If D is not a perfect square
then the roots are irrational and distinct.
2. If D = 0 then the roots are real and equal.
3. If D < 0 then the quadratic equation has no
real roots.
Example :
1.1 Find the discriminant of Quadratic Equation `x^25x+6=0` and discuss the nature of its roots
`x^25x+6=0`
`=> x^25x+6 = 0`
Comparing the given equation with the standard quadratic equation `ax^2 + bx + c = 0,`
we get, `a = 1, b = 5, c = 6.`
`:. Delta = b^2  4ac`
` = (5)^2  4 (1) (6)`
` = 25  24`
` = 1`
` = (1)^2`
Here, `Delta > 0` and is a perfect square. Also a and b are rational.
Hence, the roots of the equation are distinct and rational.
1.2 Find the discriminant of Quadratic Equation `25X^230X=+9` and discuss the nature of its roots
`25X^230X=+9`
`=> 25X^230X9 = 0`
Comparing the given equation with the standard quadratic equation `ax^2 + bx + c = 0,`
we get, `a = 25, b = 30, c = 9.`
`:. Delta = b^2  4ac`
` = (30)^2  4 (25) (9)`
` = 900 + 900`
` = 1800`
Here, `Delta > 0` but not a perfect square.
Hence, the roots of the equation are distinct and irrational.
2. Find the discriminant of Quadratic Equation `9X^224X+16=0` and discuss the nature of its roots
`9X^224X+16=0`
`=> 9X^224X+16 = 0`
Comparing the given equation with the standard quadratic equation `ax^2 + bx + c = 0,`
we get, `a = 9, b = 24, c = 16.`
`:. Delta = b^2  4ac`
` = (24)^2  4 (9) (16)`
` = 576  576`
` = 0`
Here, `Delta = 0,` the roots of the equation are real and equal.
and since a and b are both rational, the roots are rational.
Thus, the roots of the given equation are equal and rational.
3. Find the discriminant of Quadratic Equation `4X^2+11X+10=0` and discuss the nature of its roots
`4X^2+11X+10=0`
`=> 4X^2+11X+10 = 0`
Comparing the given equation with the standard quadratic equation `ax^2 + bx + c = 0,`
we get, `a = 4, b = 11, c = 10.`
`:. Delta = b^2  4ac`
` = (11)^2  4 (4) (10)`
` = 121  160`
` = 39`
Here, `Delta < 0`
Hence, the equation has no real roots.

Discriminant & Nature Of Roots

Here X^{2} = X^2 = X2 and 2X = 2*X
1. 25X^{2}  30X + 9 = 0
2. 2X^{2} + 5X  10 = 0
3. X^{2} + 10X  56 = 0
4. 4X^{2} + 11X + 10 = 0
5. X^{2}  2X = 8
6. X^{2}  25 = 0
7. X^{2} + 5X + 3 = 0
8. 9X^{2}  24X + 16 = 0



