Join Communities
Enhance your Writing Skills |
Become a Photographer |
Handigifts |
Learn about Jazz & More |
Love the Art |
QUADRATIC EXPRESSION: If are complex numbers then is called a quadratic expression in x.
QUADRATIC EQUATION: If are complex numbers then is called a quadratic equation in x.
ROOT OF A QUADRATIC EQUATION: If then is a root or solution of the quadratic equation
A quadratic equation can not have more than two roots or two solutions. The roots of are and its discrminent is .
NATURE OF THE ROOTS OF THE EQUATION
- If a,b,c are real and , then the roots are real and distinct.
- If a,b,c are real and , then the roots are real and equal.
- If a,b,c are real and , then the roots are two conjugate complex numbers.
- If a,b,c are rational and , and is a perfect square then the roots are rational and distinct.
- If a,b,c are rational and , and is not a perfect square then the roots are conjugate surds i.e .
- If a,b,c are rational and , then the roots are conjugate complex numbers i.e, .
FORMATION OF THE QUADRATIC EQUATION WITH ROOTS AND :The quadratic equation whose roots are and is .
RELATION BETWEEN THE ROOTS OF
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
PROPERTIES OF ROOTS OF THE EQUATION
If a and c are of the same sign i.e, is +ve, then both the roots are of same sign.
If a and c are of opposite sign i.e, is -ve, then the roots are of opposite sign.
If both the roots are -ve, then a,b,c will have the same sign.
If both the roots are +ve then a, c will have the same sign different from the sign of b.
If a=c, then the roots are reciprocal to each other.
If a+b+c=0,then the roots are 1 and
If a+c=b, then the roots are -1 and
If the roots are in the ratio m:n then
If one root is p times the other root then
If one root is equal to the n th power of the other root thenIf one root is square of the other, then
If roots differ by unity, then
SAME ROOTS: If and have the same roots then
ONE ROOT IS COMMON:The equations and where , have one common root then and the common root is
SIGNS OF AND
If the equation has complex roots then and will have the same sign
If the equation has equal roots then and will have same sign
If the equation has real roots then
1. and will have opposite sign.
2. or and will have same sign.
MAXIMUM OR MINIMUM VALUE O QUADRATIC EXPRESSION
If a > 0, then the minimum value of is (This value is attained at ).
If a < 0, then the maximum value of is (This value is attained at ).If are the roots of then the equation whose roots are
1.
2.
3.
4.
5.
6.
7.
LOCATING THE ROOTS OF QUADRATIC EQUATION UNDER GIVEN CONDITIONS
Both the roots of equation are greater than a given number if
Both the roots of equation are smaller than a given number if
Exactly one root of lies between the numbers if
but f(p) and f(q) are not simultaneously zero.Both the roots of equation lie between two given numbers if
The extreme values of are
Now let us get into some practice exercises
Moderate Level Practice Exercise
All time most popular tags
Post Comments