Mathematics NATURE OF ROOTS OF QUADRATIC EQUATION :

Please Wait... While Loading Full Video

NATURE OF ROOTS OF QUADRATIC EQUATION :

Natue of Roots of Quadratic Equations :

Consider the quadratic equation

`ax^2 + bx + c = 0`

where `a, b, c in R` and `a ne 0`.

Roots of the equation are given by

`x=(-b pm sqrt(b^2-4ac))/(2a)`

Now, we observe that the roots depend upon the value of the quantity `b^2 - 4ac`. This quantity is generally denoted by `D` and is known as the discriminant of the quadratic equation which decides nature of the roots. We also observe the following results :

`(i)` lf `D > 0 =>` roots are real and distinct.

`(ii)` lf `D = 0 =>` roots are equal.

`text(Note :)` From `(i)` and `(ii)` it is clear that for real roots of a quadratic equation `D` must be greater than or equal to zero. `(i.e. D >=0)`

`(iii)` If `D < 0 =>` roots are imaginary.

`text(Important Note:)`

`(1)` If co-efficients of the quadratic equation are rational then its irrational roots always occur in pair. lf `p+sqrt(q)` is one of the roots then other root will be `p -sqrt(q)`.

`(2)` If co-efficients of the quadratic equation are real then its imaginary roots always occur in complex conjugate pair. lf `p + iq ` is one of the roots then other root will be `p - iq.`

Consider the quadratic equation

`ax^2 + bx + c = 0`

where `a, b, c in R` and `a ne 0`.

Roots of the equation are given by

`x=(-b pm sqrt(b^2-4ac))/(2a)`

Now, we observe that the roots depend upon the value of the quantity `b^2 - 4ac`. This quantity is generally denoted by `D` and is known as the discriminant of the quadratic equation which decides nature of the roots. We also observe the following results :

`(i)` lf `D > 0 =>` roots are real and distinct.

`(ii)` lf `D = 0 =>` roots are equal.

`text(Note :)` From `(i)` and `(ii)` it is clear that for real roots of a quadratic equation `D` must be greater than or equal to zero. `(i.e. D >=0)`

`(iii)` If `D < 0 =>` roots are imaginary.

`text(Important Note:)`

`(1)` If co-efficients of the quadratic equation are rational then its irrational roots always occur in pair. lf `p+sqrt(q)` is one of the roots then other root will be `p -sqrt(q)`.

`(2)` If co-efficients of the quadratic equation are real then its imaginary roots always occur in complex conjugate pair. lf `p + iq ` is one of the roots then other root will be `p - iq.`

Roots Under Particular Cases :

Consider the quadratic equation
`ax^2 + bx+ c= 0...................(i)`
where `a, b, c in R` and `a ne 0.` Then, the following hold good :

(i) If roots of Eq. (i) are equal in magnitude but opposite in sign, then sum of
roots is zero as well as D > 0, i.e. b = 0 and D > 0.
(ii) If roots of Eq. (i) are reciprocal to each other, then product of roots is 1 as
well as `D >= 0` i.e., a= c and `D >= 0` .
(iii) If roots of Eq. (i) are of opposite signs, then product of roots < 0 as well as
D > 0 i.e., a> 0, c < 0 and D > 0 or a< 0, c > 0 and D > 0.
(iv) If both roots of Eq. (i) are positive, then sum and product of roots > 0 as
well as `D >= 0` i.e., a> 0, b< 0, c> 0 and `D >= 0` or a < 0, b> 0, c < 0 and `D >= 0.`
(v) If both roots of Eq. (i) are negative, then sum of roots< 0, product of roots> 0
as well as `D >= 0` i.e., a > 0 , b>0 ,c>0 and `D>=0` or `a<0` , `b<0` , `c<0` and
`D>=0`.
(vi) If atleast one root of Eq. (i) is positive, then either one root is positive or
both roots are positive i.e., point (iii) `uu` (iv).
(vii) If atleast one root of Eq. (i) is negative, then either one root is negative or
both roots are negative i.e., point (iii) `uu` (v).
(viii) If greater root in magnitude of Eq. (i) is positive, then
sign of b = sign of `c ne` sign of a.
(ix) If greater root in magnitude of Eq. (i) is negative, then
sign of a = sign of `b ne` sign of c.
(x) If both roots of Eq. (i) are zero, then b = c = 0.
(xi) If roots of Eq. (i) are 0 and `(-b/a)` , then c=0 ,
(xii) If roots of Eq. (i) are 1 and `c/a` , then a+ b + c = 0.

Consider the quadratic equation
`ax^2 + bx+ c= 0...................(i)`
where `a, b, c in R` and `a ne 0.` Then, the following hold good :

(i) If roots of Eq. (i) are equal in magnitude but opposite in sign, then sum of
roots is zero as well as D > 0, i.e. b = 0 and D > 0.
(ii) If roots of Eq. (i) are reciprocal to each other, then product of roots is 1 as
well as `D >= 0` i.e., a= c and `D >= 0` .
(iii) If roots of Eq. (i) are of opposite signs, then product of roots < 0 as well as
D > 0 i.e., a> 0, c < 0 and D > 0 or a< 0, c > 0 and D > 0.
(iv) If both roots of Eq. (i) are positive, then sum and product of roots > 0 as
well as `D >= 0` i.e., a> 0, b< 0, c> 0 and `D >= 0` or a < 0, b> 0, c < 0 and `D >= 0.`
(v) If both roots of Eq. (i) are negative, then sum of roots< 0, product of roots> 0
as well as `D >= 0` i.e., a > 0 , b>0 ,c>0 and `D>=0` or `a<0` , `b<0` , `c<0` and
`D>=0`.
(vi) If atleast one root of Eq. (i) is positive, then either one root is positive or
both roots are positive i.e., point (iii) `uu` (iv).
(vii) If atleast one root of Eq. (i) is negative, then either one root is negative or
both roots are negative i.e., point (iii) `uu` (v).
(viii) If greater root in magnitude of Eq. (i) is positive, then
sign of b = sign of `c ne` sign of a.
(ix) If greater root in magnitude of Eq. (i) is negative, then
sign of a = sign of `b ne` sign of c.
(x) If both roots of Eq. (i) are zero, then b = c = 0.
(xi) If roots of Eq. (i) are 0 and `(-b/a)` , then c=0 ,
(xii) If roots of Eq. (i) are 1 and `c/a` , then a+ b + c = 0.

SiteLock