**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
then

If 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 ).

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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

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