QUADRATIC EQUATION & EXPRESSION

JEE MAINS Syllabus

1. Polynomial
2. Quadratic Expression

3. Quadratic Equation
4. Solution of Quadratic Equation
5. Nature of Roots
6. Sum and Product of Roots
7. Formation of an equation with given roots
8. Roots Under Particular Cases
9. Condition for Common Roots
10. Nature of the Factors of the Quadratic Expression
11. Maximum and Minimum Value of Quadratic Expression
12. Sign of the Quadratic Expression

KEY CONCEPT
QUADRATIC EQUATION & EXPRESSION 3
 Where, a, b, c  C and a  0
3.1 Roots of a Quadratic Equation
Algebraic expression containing many terms is called
The values of variable x which satisfy the quadratic
Polynomial.
equation is called as Roots (also called solutions or
e.g 4x4 + 3x3 – 7x2 + 5x + 3, 3x3 + x2 – 3x + 5 zeros) of a Quadratic Equation.
f(x) = a0 + a1x + a2x2 + a3x3 +..... an–1 x n–1 +anxn
where x is a variable, a0,a1, a2 ..........anC.
4.1 Factorization Method :
1.1 Real Polynomial : Let a0, a1,a2.....an be real numbers
and x is a real variable. Let ax2 + bx + c = a(x–) (x –) = 0
Then f(x) = a0 + a1 x + a2 x2 +..... + an xn is called real Then x =  and x =  will satisfy the given
polynomial of real variable x with real coefficients. equation.
eg. – 3x3 – 4x2 + 5x – 4, x 2– 2x + 1 etc. are real Hence factorize the equation and equating each to zero
polynomials. gives roots of equation.
1.2 Complex Polynomial: If a0,a1,a2...an be complex e.g. 3x2 – 2x –1 = 0 (x – 1)( 3x + 1) = 0
numbers and x is a varying complex number, then
f(x) = a0+ a1x + a2x2 +...... anxn is called a complex x =1,
polynomial of complex variable x with complex
4.2 Hindu Method (Sri Dharacharya Method) :
coefficients.
By completing the perfect square as
eg.- 3x2 – (2+ 4 i) x + (5i–4), x3 –5ix2 +
(1+2i) x+4 etc. are complex polynomials.
ax2 + bx + c = 0  x2 + x+ =0
1.3 Degree of Polynomial : Highest Power of variable x
in a polynomial is called as a degree of polynomial.
Adding and substracting
e.g. f(x)=a0+a1x+a2x2+a3x3+....an–1xn–1+anxn is n degree
polynomial.
f(x) = 4x3 + 3x2 – 7x + 5 is 3 degree polynomial  =0
f(x) = 3x – 4 is single degree polynomial or Linear
polynomial.
Which gives, x =
f(x) = bx is odd Linear polynomial
Hence the Quadratic equation ax 2 + bx + c = 0
(a  0) has two roots, given by
A Polynomial of degree two of the form
ax2 + bx + c (a  0 ) is called a quadratic expression in = and  =
x.
Note : Every quadratic equation has two and only two
e.g 3x2 + 7x + 5, x2 – 7x + 3
roots.
General form : – f(x) = ax2 + bx + c
where a, b, c  C and a  0
In Quadratic equation ax2 + bx + c = 0, the term
b2 – 4ac is called discriminant of the equation, which
plays an important role in finding the nature of the
A quadratic Polynomial f(x) when equated to zero is
called Quadratic Equation. roots. It is denoted by  or D.

e.g 3x2 + 7x + 5 = 0, – 9x2 + 7x + 5 = 0, (A) Suppose a, b, c  R and a  0 then

2
x + 2x = 0, 2
2x = 0 (i) If D > 0  Roots are Real and unequal
General form : (ii) If D = 0 Roots are Real and equal and
each equal to –b/2a
ax2 + bx + c = 0
QUADRATIC EQUATION & EXPRESSION 4
 (iii) If D < 0  Roots are imaginary and unequal or (v) 3 – 3 = ( – )3 + 3( – )
complex conjugate.
=
(B) Suppose a, b, c  Q, a  0 then
(i) If D > 0 and D is perfect square =
 Roots are unequal and Rational
(ii) If D > 0 and D is not perfect square (vi) 4 + 4 = –222
 Roots are irrational and unequal
5.1 Conjugate Roots : = –2
The Irrational and complex roots of a quadratic equation
are always occurs in pairs. Therefore (a, b, c,Q) (vii)4 – 4 = (2 – 2) (2 + 2)
If One Root then Other Root
 + i  – i =
+ –
(viii)2 + + 2 = ( + )2 – 

(ix) + = =
If  and  are the roots of quadratic equation
ax2 + bx + c = 0, then,
(i) Sum of Roots (x) 2 + 2 = ( + )

S=+=– =– (xi) + = =

(ii) Product of Roots

P = = =
A quadratic equation whose roots are  and  is given
by
e.g. In equation
3x2 + 4x – 5 = 0 (x – ) (x – ) =0
 x2 – x – x +  = 0
Sum of roots S =– ,
 x2 – ( + )x +  = 0
Product of roots P =– i.e x2 – (sum of Roots)x + Product of Roots = 0

6.1 Relation between Roots and Coefficients

 x2 – Sx + P = 0
If roots of quadratic equation ax2 + bx + c = 0 (a  0) 7.1 Equation in terms of the Roots of another Equation
are and then : If are roots of the equation ax 2 + bx + c = 0 then the
equation whose roots are
(i) ( – ) = = ± =
(i) –, –ax2 – bx + c = 0
(Replace x by – x)
(ii) 1/, 1/cx2 + bx + a = 0
(Replace x by 1/x)
(ii)  +  = ( + ) – 2 =
2 2 2 (iii)  ,  ; n  N a(x ) + b (x ) + c = 0
n n 1/n 2 1/n

(Replace x by x1/n)
(iii) 2 – 2 = ( + ) (iv) k, k ax2 + kbx + k2 c = 0
(Replace x by x/k)
=– = (v) k + , k +   a(x– k)2 + b (x – k) + c = 0
(Replace x by (x–k))

(iv) 3 + 3 = ( + )3 – 3( + ) = –

QUADRATIC EQUATION & EXPRESSION 5

 9.1 Only One Root is Common : Let  be the common
(vi) , k2 ax2 + kbx + c = 0 root of quadratic equations
(Replace x by kx) a1x2 + b1x + c1 = 0 and a2x2 + b2x + c2 = 0 then
(vii)  ,  ; n  N  a(x ) + b(x ) + c = 0
1/n 1/n n 2 n
 a12 + b1 + c1 = 0
(Replace x by xn)
a22 + b2 + c2 = 0
7.2 Symmetric Expressions
The symmetric expressions of the roots , of an By Cramer's rule :
equation are those expressions in and , which do not
change by interchanging and . To find the value of
such an expression, we generally express that in terms = =
of and .
Some examples of symmetric expressions are–
or
(i) 2 + 2 (ii) 2 + + 2
= =
(iii) + (iv) +

(v) 2 + 2 (vi) + = , 2 = ,  0.

(vii) 3 + 3 (viii) 4 + 4
 The condition for only one Root common is (c 1a2
– c2a1)2 = (b1c2 – b2c1)(a1b2 – a2b1)
9.2 Both roots are common : Then required conditions is
For the quadratic equation ax2 + bx + c = 0
(i) If b = 0  roots are of equal = =
magnitude but of
opposite sign
Note : Two different quadratic equation with rational
(ii) If c = 0  one root is zero other coefficient cannot have single common root
is – b/a which is complex or irrational, as imaginary and
(iii) If b = c = 0  both root are zero surd roots always occur in pair.
(iv) If a = c  roots are reciprocal to
each other

(v) If Roots are of opposite signs

The nature of factors of the quadratic expression
(vi) If Both roots are negative. ax2 + bx + c is the same as the nature of
roots of the corresponding quadratic equation
ax2 + bx + c = 0 (a, b, c, R). Thus the factors of the
(vii)  Both roots are positive. expression are:
(i) Real and different, if b2 – 4ac > 0.
(viii) If sign of a = sign of b  sign of c
(ii) Rational and different, if b2 – 4 ac is a perfect
Greater root in magnitude is negative. square where (a, b, c, Q).
(ix) If sign of b = sign of c  sign of a (iii) Real and equal , if b2 – 4 ac = 0.
Greater root in magnitude is positive.
(iv) Imaginary, if b2 – 4 ac < 0.
(x) If a + b + c = 0 one root is 1 and second
root is c/a. eg. The factors of x2 – x + 1 are -
(xi) If a = b = c = 0 then equation will become an Sol. The factors of x2 – x + 1 are imaginary because
identity and will be satisfy by every value of x.
b2 – 4 ac = (–1)2 – 4(1) (1)
= 1 – 4 = –3 < 0

QUADRATIC EQUATION & EXPRESSION 6

 (i) ax2 + bx + c > 0 has a solution any x  –
if a > 0 and has no solution if a < 0;
2
In a Quadratic Expression ax + bx + c
(ii) ax2 + bx + c < 0 has a solution any x 
(i) If a > 0 Quadratic expression has least value at
if a < 0 and has no solution if a > 0;
x=– . This least value is given by
(iii) ax2 + bx + c  0 has any x as a solution
if a > 0 and the unique solution
=–
x= , if a < 0;
(ii) If a < 0, Quadratic expression has greatest value s
at x = . This greatest value is given by (iv) ax2 + bx + c  0 has any x as a solution

if a < 0 and x = , if a > 0;

=–
Case 3.
D < 0 then from (1)
(i) if a > 0, then ax2 + bx + c > 0 for all x;
Let y = ax2 + bx + c (a  0) (ii) if a < 0, then ax2 + bx + c < 0 for all x.
eg. The sign of x2 + 2x + 3 is positive for all
y=a
x R, because here
b2 –4 ac = 4 – 12 = –8 < 0 and a = 1 > 0.
=a eg. The sign of 3x2 + 5 x – 8 is negative for all
x R because here
b2 –4 ac = 25 – 96 = – 71<0 and a = – 3 < 0
=a ...(1) 12.1 Graph of Quadratic Expression :
Consider the expression y = ax 2 + bx + c,
Where D = b2 – 4ac is the Discriminant of the a  0 and a,b,c  R then the graph between x, y is
quadratic equation ax2 + bx + c = 0 always a parabola. If a > 0 then the shape of the
Case 1. parabola in concave upward and if a < 0 then the
D > 0 : Suppose the roots of ax 2 + bx + c = 0 are shape of the parabola is concave downwards.
 and  and  >  (say). There is only 6 possible graph of a Quadratic
,  are real and distinct. expression as given below :
Then ax2 + bx + c = a(x –)(x – ) Case - I When a > 0
Clearly (x –)(x – ) > 0 for x <  and x <  (i) If D > 0
since both factors are of the same sign and Roots are real and different ( X1 and X2)
(x –)(x – ) < 0 for  > x > 
Minimum value LM = at
For x =  or x =  , (x –)(x – ) = 0
 If a > 0, then ax2 + bx + c > 0 for all x outside the x = OL = – b/2a
interval [, ] and is negative for all x in (, ).
y is positive for all x out side interval [x1, x2 ]
If a < 0, then its viceversa.
and is negative for all x inside (x1, x2)
Case 2.
D = 0 then from (1)

ax2 + bx + c =

 x – , the quadratic expression takes

on values of the same sign as a;
If x = – b/2a then ax2 + bx + c = 0.
 If D = 0, then (ii) If D = 0
QUADRATIC EQUATION & EXPRESSION 7
 Roots are equal (OA) (iii) If D < 0
Min. value = 0 at x = OA = –b/2a Roots are complex conjugate
y > 0 for all x  y is negative for all x  R

(iii) If D < 0
Roots are complex conjugate The general form of a quadratic expression in two
y is positive for all x R. variable x & y is
ax2 + 2hxy + by2 + 2 gx + 2 fy + c
The condition that this expression may be resolved
into two linear rational factors is

 = =0

abc + 2 fgh – af2 – bg2 – ch2 = 0 and h2 – ab > 0

Case- II When a < 0
This expression is called discriminant of the above
(i) If D > 0
quadratic expression.
Roots are real and different ( x1 and x2)

Max. value = LM = at x = OL =
(i) Every equation of nth degree (n  1 ) has exactly n
–b/2a roots and if the equation has more than n roots, it
y is positive for all x inside ( x1, x2 ) and is an identity.
y is negative for all x outside [ x1, x2] (ii) If  is a root of the equation f (x) = 0 then the
polynomial f (x) is exactly divisible by
(x–) or (x – ) is a factor of f (x)
(iii) If quadratic equations a1 x2 + b1 x+c1 = 0 and
a2 x2 + b2 x + c2 = 0 are in the same ratio

then

(ii) If D = 0 =
Roots are equal = OA
Max. value = 0 at x = OA = –b/2a (iv) If one root is k times the other root of quadratic
equation a1 x2 + b1 x+ c1 = 0 then
y is negative for all x 
=

(v) Quadratic equations containing modulas sign are

solved considering both positive and negative
values of the quantity containing modulus sign.
Finally the roots of the given equation will be
those values among the values of the variable so
obtained which satisfy the given equation.

QUADRATIC EQUATION & EXPRESSION 8