1) D 2 Roots of the Quadratic Equation Vidyalankar

Std.X.IIT 3 yrs/Foundation/Maths/Ch.1/Pg.1

IIT JEE 3 Years Course Mathematics : Problems on Quadratic Equations and Expressions

Theory

An equation of the form : ax2 + bx + c = 0 … (1)

where a, b, c are real numbers and a 0 is called a quadratic equation. The numbers

a, b, c are called the coefficients of the quadratic equation. The quantity D (D = b2 4ac) is

known as the discriminant of the equation. A root of a quadratic equation is a number (real

or complex) such that a2 + b + c = 0.

Roots of the Quadratic Equation

Consider any quadratic equation in standard form:

ax2 + bx + c = 0

ax2 + bx = c

2 bx cx

a a

Half of b

a is

2 2

2

b b band

2a 2a 4a

.

Thus add 2

2

b

4a on both sides,

2 2

2

2

bx b c bx

a 2a a 4a

2 2

2

b b 4acx

2a 4a

2b b 4ac

x2a 2a

x =

2b b 4ac

2a

.

So, if and are the roots of ax2 + bx + c = 0 then

=b D b D

,2a 2a

.

The quantity D (D = b2 4ac) is known as the discriminant of the equation.

Nature of Roots

1. If a, b, c are real numbers (a, b, c R) and a 0, then

a) If D < 0, equation (1) has non real complex roots.

b) If D > 0, equation (1) has real and distinct roots, namely

=b D

2a

, =

b D

2a

and then ax2 + bx + c = a(x ) (x ) …(2)

c) If D = 0, then equation (1) has real and equal roots.

= = b

2a

and then ax2 + bx + c = a (x )2 = a(x )2 …(3)

To represent the quadratic ax2 + bx + c in the form (2) and (3) is to expand it into

linear factors.

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(2) Vidyalankar : Foundation Mathematics


2. If a, b, c Q (Q = rational numbers) and D is a perfect square, then the roots are rational

and in case it be not a perfect square then the roots are irrational.

3. If a, b, c R and p + iq is one root of equation (1) (q 0) then the other root must be the

conjugate p iq and viceversa. (p, q R and i = 1 ).

4. If a, b, c Q and p + q is one root of equation (1) then the other root must be the

conjugate p q and viceversa.

5. If a = 1 and b, c I (I = integer) and the roots of equation (1) are rational numbers, then

these roots must be integers.

6. If the quadratic equation is satisfied by more than two distinct numbers (real or complex),

then it becomes an identity, i.e., a = b = c = 0.

Some Important Forms

i) An equation of the form

(x a) (x b) (x c) (x d) = A

where a < b < c < d, b a = d c, can be solved.

be a change of variable

i.e., y = (x a) (x b) (x c) (x d)

4

or y = x

(a b c d)

4

ii) An equation of the form

(x a) (x b) (x c) (x d) = 2Ax

where ab = cd, can be reduced to a collection of two quadratic equations by a change of

variable y = x + ab/x.

iii) An equation of the form

4 4(x a) (x b) A

can also be solved by a change of variable, i.e., making a substitution.

y = (x a) (x b)

2

Sum and Product of Roots

If and are the roots of equation (1), then

i) Sum of roots = + = b/a

ii) Product of roots = = c/a

iii) The quadratic equation whose roots are and is

(x ) (x ) = 0

i.e., x2 ( + ) x + = 0 …(4)

Symmetric functions of the roots

If and are the roots of ax2 + bx + c = 0, then

i) + = b

a and =

c

a ii) a + b =

c

and a + b =

c

iii) 2 2 2( ) 2 iv) = 2( ) 4

v) 2 2 ( )( ) vi) 3 3 3( ) 3 ( )

vii) 4 4 2 2 2 2 2( ) 2

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Problems on Quadratic Equations and Expressions (3)


Condition for Common roots

Equations: 21 1 1a x b x c 0

22 2 2a x b x c 0

Roots: 1, 1 and 2, 2.

Condition for one common root :

21 2 2 1 1 2 2 1 1 2 2 1(c a c a ) (b c b c )(a b a b )

Condition for both the roots to be common:

1 1 1

2 2 2

a b c

a b c

Note: If the coefficient of x2 in both the equations be unity, then the value of common

root is obtained by subtracting their equations.

Identity

f(x) = 0 is said to be an identity in x if it is satisfied by all values of x in the domain of f(x).

To verify whether an equation is an identity, put (n + 1) values (where n is the highest

power of x) in the given relation. If the given relation is satisfied by all these values of x,

then it is an identity.

Absolute value function (or Modulus function) :

The absolute value function is written as f(x) = | x | as

is given by the formula

x ; x 0f(x)

x ; x 0

Properties of modulus functions :

i) | x | a a x a; (a 0)

ii) | x | a x a and x a; (a 0)

iii) | | x | | y || | x y | | x | + | y |

iv) | x + y | = | x | + | y | if x and y are of same sign xy 0.

Solved Examples

1. Solve the equation 2x x5 24.5 25 0 .

Solution :

Let 5x = t, then the given equation can reduce in the form

2t 24t 25 0

(t 25) (t + 1 ) = 0

t 1, t = 25

then 5x = 25

then x1 = 2

Hence, x1 = 2 is only one root of the original equation.

x

|x|

0

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2. Determine value(s) of p for which the quadratic equation 2x2 + px + 8 = 0 has real roots.

Solution :

D = b2 4ac = p2 (4 2 8) = p2 64

For the equation having real roots, D 0,

i.e., p2 64 0 p2 64 p2 82.

This gives p 8 or p 8.

3. The number of real solutions of the equation 5 5

x 5x 5 x 5

; is

(A) 1 (B) 2 (C) 0 (D)

Solution :

The given equation is meaningful for x 5. But for x 5, it reduces to x = 5. Thus, the

given equation has no real solution. Hence (C).

4. If , be the roots of 2x px q 0 , form the equation whose roots are ( )2 and ( + )2.

Solution :

It is given that , are roots of 2x px q 0

+ = p and = q ….(1)

Let = ( )2 and = ( + )2

+ = 2 2 2 22

= 2 22 2 2 p 2q

from (1)

= 2 2 2

= 2

2 2 2 24 = 2

2 22 2

= 22 2 2 2p 2q 2q p p 4q

The equation whose roots are and is

2x x 0

i.e. 2 2 2 2x 2 p 2q x p p 4q 0

5. Solve for real x : (x2 + x) (x2 + x 2) = 24.

Solution :

Put x2 + x = m Hence given equation becomes m (m 2) = 24 m2 2m = 24

m2 2m 24 = 0 (m 6) (m + 4) = 0 m = 6 or m = 4

x2 + x = 6 … (I) or x2 + x = 4 … (II)

From (I), x2 + x 6 = 0

(x + 3) (x 2) = 0 x = 3 or x = 2

From (II), x2 + x + 4 = 0

x = 1 1 16

2

=

1 15

2

These roots are not real and so not admissible.

The solution set of the given equation = {3, 2}.

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6. If a, b are the roots of x2 + px + q = 0 and c, d are the roots of x2 px + r = 0 then

a2 + b2 + c2 + d2 =

(A) p2 q r (B) 2 (p2 q r) (C) p2 + q + r (D) p2 + q2 r2

Solution :

a2 + b2 + c2 + d2

(a + b)2 2ab + (c + d)2 2cd = p2 2q + p2 2r

= 2 (p2 q r). Hence (B).

7. If the roots of 2ax bx c 0 are sin and cos for some then,

(A) a2 + b2 = 2ac (B) a2 b2 = 2ac (C) b2 a2 = 2ac (D) none of these

Solution :

2 2sin cos 1

2(sin cos ) 2sin cos 1

2

2

b c2 1

aa

2 2b 2ac a 2 2a b 2ac Hence (B).

8. If (a2 1) x2 + (a 1) x + a2 4a + 3 = 0 is an identity in x, then find the value/s of ‘a’.

Solution :

The given equation is an identity, hence it is satisfied by all real values of x.

Therefore, all the coefficients must be zero.

a2 1 = 0 a2 = 1 a = 1

a 1 = 0 a = 1

a2 4a + 3 = 0 (a 3) (a 1) = 0 a = 1, 3

Hence, common value of a = 1.

9. If the equations 2x2 + x + = 0 and x2 + 2x + 2 = 0 have a common root, then find the

real values of .

Solution :

Equations having common root are

2x2 + x + = 0 and

x2 + 2x + 2 = 0

(4 )2 = (4 ) (22 2)

(3)2 = 82 23 8 + 22

92 = 23 + 102 8

23 2 + 8 = 0

(22 + 8) = 0

= 0 or 22 + 8 = 0

We find that (22 + 8 = 0) has no real roots. Hence, only real value of is = 0

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10. The equations x3 + 4x2 + px + q = 0 and x3 + 6x2 + px + r = 0 have two common roots,

where p, q, r R. Then find the equation formed by their uncommon roots.

Solution :

Let , , be roots of equation x3 + 4x2 + px + q = 0 … (i)

and , , be roots of equation x3 + 6x2 + px + r = 0 … (ii)

equation (i) equation (ii) gives us,

2x2 + q r = 0 This equation has roots , .

2x2 + 0x + (q r) = 0

+ = 0

2= 0 … (iii)

We know that if x1, x2, x3 are root of equation (ax3 + bx2 + cx + d = 0), then

1 2 3

bx x x

a

Using above property, we write

+ + = 4 for equation (i) and + + = 6 for equation (ii)

Substituting value of ( + ) from (iii) into above equations, we get

0 + = 4 = 4 and 0 + = 6 = 6

Hence, required equation having roots as 4 and 6 is

(x + 4) (x + 6) = 0 x2 + 10x + 24 = 0

11. The sum of all the real roots of the equation 2

x 3 x 2 1 0 is

(A) 12 (B) 10 (C) 5 (D) 3

Solution : (C)

2(x 6x 9) x 2 1 0.

If x 2: 2x 6x 9 x 2 1 0

2x 5x 6 0

(x 3)(x 2) 0

x 3 or x 2

If x 2: 2x 6x 9 2 x 1 0

2x 7x 10 0

(x 5)(x 2) 0

x 2 or x 5

But not solution

Sum of roots 3 2 5

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EXERCISE PROBLEM

1. The number of solutions of the equation x x xsin(e ) 5 5 is :

(1) 0 (2) 2 (3) 1 (4) infinitely many

2. The number of real roots of

2| x | 3 | x | 2 0 is :

(1) 0 (2) 2 (3) 4 (4) 1

3. The value of 6 6 6 .......to is :

(1) 3 (2) 6 (3) 2 (4) 4

4. The real roots of the equation 2

7log (x 4x 5)7 x 1,

are :

(1) 1 and 2 (2) 2 and 3 (3) 3 and 4 (4) 4 and 5

5. The set of all real numbers ‘x’ for which 2x | x 2 | x 0, is

(1) (∞, 2) (2, ∞) (2) ( , 2) ( 2, )

(3) (∞, 1) (1, ∞) (4) ( 2, )

6. If the equations 2x ax b 0 and 2x a 'x b ' 0 have a common root, then this root

must be equal to :

(1) b b '

a a '

(2)

b b '

a ' a

(3)

b b '

a a '

(4)

b b '

a ' a

7. If , are the roots of 2ax bx c 0 , then roots of 2cx bx a 0 are :

(1) , (2) 1

,

(3) , 1

(4)

1 1,

8. If , are the roots of the equation 2x x 0 , then the values of and are :

(1) = 1, = 1 (2) = 1, = 2 (3) = 2, = 1 (4) = 2, = 2

9. The maximum value of 25 4x 4x is :

(1) 6 (2) 7 (3) 2 (4) 3

10. If the equations k 2 2(6x 3) rx 2x 1 0 and 2 26k(2x 1) px 4x 2 0 have

both roots common, then the value of (2r p) is :

(1) 0 (2) 1

2 (3) 1 (4) none of these

11. If sin and cos are roots of 2px qx r 0 , then :

(1) 2 2p q 2pr 0 (2) 2 2 2(p r) q r (3) 2 2p q 2pr 0 (4) none of these

12. If c, d are roots of (x a) (x b) k = 0, then a, b are roots of the equation :

(1) (x c) (x d) k = 0 (2) (x c) (x d) + k = 0

(3) (x a) (x c) + k = 0 (4) (x b) (x d) + k = 0

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13. If x + 1 is a factor of 4 3 2x (p 3)x (3p 5)x (2p 9)x 6 , then the value of p is :

(1) 4 (2) 0 (3) 4 (4) 2

14. If 2x ax b 0 and 2x bx a 0 have a common root, then the numerical value of

a + b is :

(1) 1 (2) 0 (3) 1 (4) none of these

15. If and are the roots of 2x px 1 0 and , the roots of 2x qx 1 0, then the

value of ( ) ( ) ( + ) ( + ) is :

(1) p2 q2 (2) q2 p2 (3) p2 (4) q2

16. The value of ‘a’ for which the equation 2 2 23x 2(a 1)x (a 3a 2) = 0 possesses

roots of opposite sign lies in :

(1) (, 1) (2) (, 0) (3) (1, 2) (4) (3/2, 2)

17. If a, b, c are positive terms of an A.P., then the roots of 2ax 2bx c 0 are

(1) real and distinct (2) real and equal (3) imaginary (4) none of these

18. If the roots of a(b c) x2 + b(c a) x + c (a b) = 0 are equal, then a, b, c are in

(1) A.P. (2) G.P. (3) H.P. (4) none of these

19. If ‘x’ satisfies |x 1| + |x 2| + |x 3| 6, then:

(1) 0 x 4 (2) x 2 or x 4 (3) x 0 or x 4 (4) None of these.

20. The roots of 2| x x 6 | x 2 are:

(1) 2, 1, 4 (2) 0, 2, 4 (3) 0, 1, 4 (4) 2, 2, 4

21. The roots of the equation 2| 2x 1| 3 | 2x 1| 2 0 are

(1) 1 1

,0,2 2

(2) 1 3

,0,2 2

(3) 3 1

, ,0,12 2

(4) 1 3

,0,1,2 2

22. If 2x 3x 5 0 and 2ax bx 2 0 have a common root and a, b R, then

minimum value of a + b is equal to

(1) 8 (2) 4

5 (3)

8

5 (4) None of these.

23. If ‘’, ‘’ are roots of 2ax bx c 0 and + , + are roots of 2px qx r 0,

then

2

2

b 4ac

q 4pr is equal to

(1) 2(p/a) (2) 1 (3) 2(a/p) (4) 0

24. If x2 + px + 1 is a factor of the expression ax3 + bx2 + c then

(1) a2 + c2 = ab (2) a2 c2 = ab (3) a2 c2 = bc (4) None of these

25. If the roots of the equation x2 5x + 16 = 0 are , and the roots of the equation

x2 + px + q = 0 are (2 + 2) and /2, then

(1) p = 1 and q = 56 (2) q = 1 and q = 56 (3) p = 1 and q = 56 (4) p = 1 and q = 56

26. If (1 p) is a root of quadratic equation 2x + px + (1 p) = 0 then its roots are

(1) 0, 1 (2) 1, 2 (3) 0, 1 (4) 1, 1

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27. If the difference between the roots of the equation x2 + ax + 1 = 0 is less than 5, then the

set of possible values of a is

(1) (3, 3) (2) (3, ) (3) (3, ) (4) (, 3)

28. If , , , be the roots of 4 2x 6x 5x 4 0 , then

( + + ) ( + + ) ( + + ) ( + + ) is

(1) 6 (2) 5 (3) 4 (4) none of these

29. If the ratio of the roots of ax2 + bx + c = 0 be m and n then m n

n m =

(1) b

ac (2)

b

ca (3)

a

bc (4) none of these

30. If the roots of the equation 2x bx

ax c

are such that + b = 0, then the value of ‘’

is:

(1) a b

a b

(2) c (3)

1

c (4)

a b

a b

31. The number of real solutions of the equation 2x 3| x | 2 0 is

(1) 3 (2) 2 (3) 4 (4) 1

32. If 5x2 x + 6 = 0,then value of x will be

(1) 6/5, 1 (2) 6/5, 1 (3) 6/5, 1 (4) none of these

33. If a,b,c,d R , then the equation 2 2 2x ax 3b x cx b x dx 2b 0 has

(1) 6 real roots (2) at least 2 real roots

(3) 4 real roots (4) 3 real roots

34. Solve the equation 4 4(6 x) (8 x) 16 .

35. If , be the roots of 2x p x 1 c 0 , then show that 1 1 1 c and find the

value of 2 2

2 2

2 1 2 1

2 c 2 c

.

36. Solve | 3x 1 | = 2x, x R.

37. Solve | x + 4 | = 3x 2, x R.

38. Solve | x2 + x | 5 = 0 for real x.

39. Solve | x2 2x 8 | = 2x, x R.

40. Solve | x2 x 6| = x + 2, x R.

41. Solve | x2 + x + 1| = 3, x R

42. Solve 2 |x + 1|2 |x + 1| = 3, x R.

43. Solve 2 |x + 1| + |x 3| = 4, x R.

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44. Solve |x| + 2 |x 1| 3 |x 2| = 4, x R.

45. Solve 2 |3 x| + 3 |x 1| = 4, x R.

46. Solve |2x + 1| 2 |x 2| = 5, x R.

47. Solve | x2 x | + |x + 1| = 5, x R.

48. Solve |x2 + x + 1| + |2x 3| = 4x, x R.

49. Solve |2x 1| + |2x + 1| = 2, x R.

50. Solve |2x + 1 1| + |2x + 1 + 1| = 2|x + 1|.

51. Solve (a) 2x x 2

x 1

(b) 2|y| |2y 1 1| = 2y 1 + 1

52. The roots of the equation 2x 15(a b) +

2x 15(a b) = 2a, where a2 b = 1, are

(A) 2, 3 (B) 4, 14 (C) 3, 5 (D) 6, 20

53. Set of real values of x satisfying 2 2x 3 x 3(5 2 6 ) (5 2 6 ) 10 is

(A) 2, 3 (B) 2 (C) 2, 2 (D) None of these

54. The biquadratic x4 2x3 3x2 + 2x + 2 = 0 has

(A) only one natural solution (B) two integral solutions

(C) two real and two non-real solutions (D) 3 1 is largest solution

55. Prove that if the roots of 29x 4ax 4 0 are imaginary, then ‘a’ must lie between 3

and 3.

1. (1) 2. (1) 3. (1) 4. (2) 5. (2)

6. (4) 7. (4) 8. (2) 9. (1) 10. (1)

11. (1) 12. (2) 13. (3) 14. (3) 15. (2)

16. (3) 17. (1) 18. (3) 19. (3) 20. (4)

21. (4) 22. (3) 23. (3) 24. (3) 25. (4)

26. (3) 27. (1) 28. (4) 29. (1) 30. (1)

31. (2) 32. (A) 33. (B) 34. 6, 8 35. [1.00]

36. 1

,5

1 37. 3 38. 1 21

2

39. 2 2, 2(1 3)

40. 2, 2, 4 41. 1, 2 42. 1 5

,2 2

43. 1 44. [2, + ]

45. 1 46. [2, + ] 47. 1 7, 2 48. 2, 1 49. x 0

50. 2 51. (a) 1, 1, 2 (b) y 1, y = 1 52. (B) 53. (C)

54. (A),(B),(D)

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1) D 2 Roots of the Quadratic Equation Vidyalankar

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