Quadratic Equations

01 02
Basic Results Quick Look
The quantity (D = b2 – 4ac) is known as the discriminat of the Let a and b be the roots of a quadratic equation ax2 + bx + c = 0 then
quadratic equation.
• The quadratic equation has real and equal roots if and only if D b 2 − 2ac
1. a 2 + β2 = (α + β) 2 − 2αβ =
= 0, i.e., b2 – 4ac = 0 a2
• The quadratic equation has real and distinct roots if and only if D b b 2 − 4ac
> 0, i.e., b2 – 4ac > 0 2. a 2 − β2 = (α + β) (α + β) 2 − 4αβ = −
a2
• The quadratic equation has complex root with non-zero imaginary
parts if and only if D < 0, i.e., b2 – 4ac < 0 b ( b 2 − 3ac )
3. a 3 + β3 = (α + β)3 − 3αβ(α + β) = −
• If p + iq (p and q being real) is a roots of the quadratic equation a3
where i= −1 , then p – iq is also a root of the quadratic equation. 4. a 3 − β3 = (α − β)3 + 3αβ(α − β)
• If p + q is an irrational root of the quadratic equation, then  b 2 − 2ac  c2
2

{ }
2
5. a 4 + β4 = ( α + β) 2
− 2 αβ − 2 a 2 2
β =   − 2
p − q is also a root of the quadratic equation provided that all 2
a2
 a 
the coefficients are rational
±b ( b 2 − ac ) b 2 − 4ac
• The quadratic equation has rational roots if D is a perfect square 6. a 4 − β=
4
( a 2 − β2 )( a 2 + β=
2
) a4
and a, b, c are rational
• If a = 1 and b, c are integers and the roots of the quadratic equation b 2 − ac
7. α 2 + αβ + β2 = (α + β) 2 − αβ =
are rational, then the roots must be integers. a2
• If the quadratic equation is satisfied by more than two numbers α β α 2 + β2 (α + β) 2 − 2αβ b 2 − ac
(real or complex), then it becomes an identity, i.e., a = b = c = 0. 8. =
+ = =
β α αβ αβ a2
bc
9. α 2β + β2 α = αβ(α + β) = − 2
a
(α + β 2 ) − 2 a 2β 2
2 2 2
2
 α   β  α 4 + β4
10.   +   = =
β α α 2β 2 a 2 + β2
03 05
Formation of an Equation with Given Roots Condition for Common Roots
A quadratic equation whose roots are a and b is given by 1. Condition for Two Quadratic Equations to have one Common
(x – a)(x – b) = 0 Root
∴ x2 – Sx + P = 0 If ax2 + bx + c = 0 and dx2 + ex + f = 0 have a common root a(say).
i.e., x2 – (sum of roots)x + (product of roots) = 0 Then (dc – af )2 = (bf – ce)(ae – bd),
∴ x2 – (a + b)x + ab = 0 2. Both roots are common
The required condtion is a= 1 b1 c1 .
=
a2 b2 c2

04
Equation in terms of the roots of another equations

If a, b are roots of the equation ax2 + bx + c = 0, then the equation 06

whose roots are Quadratic Expression
• −α, −β ⇒ = ax 2 − bx + c 0 (Replace x by − x)
1 1  1 The expression ax2 + bx + c is said to be a real quadratic expression
• , ⇒ cx 2 + bx + a =0  Replace x by  in x where a, b, c are real and a ≠ 0. Let f (x) = ax2 + bx + c where a,
α β  x
2 b, c, ∈ R (a ≠ 0). f (x) cab be written as
n n  1  1  1

• α β , n ∈ N ⇒ α  x n  + b  x n  + c =0  Replace x by x n 
       b  4ac − b 2 
2
 b 
2
D 
f ( x) = a  x +  + 2  = a  x +  − 2  , where D =
 x  2a  4a   2a  4a 
• k α, k β ⇒ ax 2 + kbx + k 2 c =0  Replace x by 
 k b2 – 4ac is discriminant of the quadratic expression. Then y = f (x)
α β
• , ⇒ k 2 ax 2 + kbx + c = 0 (Replace x by kx) represents a parabola whose axis is parallel to the y – axis, with vertex
k k
• k ) 2 + b( x − k ) + c 0 (Replace x by ( x − k ) )
k + α, k + β ⇒ a ( x − =  b D
at A  − , −  .
1 1  2a 4a 
α n , β n ; n ∈ N ⇒ α ( x n ) + b ( x n ) + c =0 (Replace x by x n )
2 2
•
07 08

Maximum and minimum values of quadratic expression Sign of f (x)

Maximum and minimum value of quadratic expression can be found Depending on the sign of a and b2 – 4ac, f (x) may be positive, negative
out by two methods: or zero, This given rise to the following cases :

a>0 1. a > 0 and b2 – 4ac < 0 ⇔ f (x) > 0 ∀ x ∈ R

vertex
vertex
x′ x
a<0 In this case the parabola always remains above the x-axis.

1. Discriminant method: 2. a > 0 and b2 – 4ac > 0. Let f (x) = 0 have two real roots a and b
In a quadratic expression ax2 + bx + c. (a < b).
2. Vertex of the parabola Y = aX 2 is X = 0, Y = 0.

b D b
i.e., x + 0, y +
= 0⇒ x =
= − ,y=
− D / 4a x′ a b x
2a 4a 2a
The f (x) > 0 ∀ x ∈ (–∞, a) ∪ (b, ∞), and f (x) < 0 ∀ x ∈ (a, b).
 b D
Hence, vertex of y = ax2 + bx + c is  − , −  .
 2a 4a  3. a < 0 and b2 – 4ac < 0 ⇔ f (x) < 0 ∀ x ∈ R.
x′ x
b
• For a > 0, f (x) has least value at x = − .
2a
 b  D In this case the parabola always remains below the x-axis.
This least value is given by f  −  =− .
 2a  4a
4. a < 0 and b2 – 4ac = 0 ⇔ f (x) ≤ 0 ∀ x ∈ R.
b
• For a < 0, f (x) has greatest value at x = − . x′ x
2a
 b  D
This greatest value is given by f  −  =− .
 2a  4a
In this case the parabola touches the x-axis and lies below the
x-axis.
 5. a < 0 and b2 – 4ac > 0
a b
x′ x

Let f (x) = 0 have two real roots a and b (a < b). Then f (x) < 0 ∀ x ∈ (–∞, a) ∪ (b, ∞) and f (x) > 0 ∀ x ∈ (a, b).

09
Location of Roots

For location of roots, we want the roots of the equation

ax2 + bx + c = 0
to lie in a given interval. For this we impose conditions on a, b and c.

1. Both the roots are positive i.e., they lie in (0, ∞), iff roots are real, the sum of the roots as well as the product of the roots is positive.
−b c
i.e., α + β= > 0 and αβ= > 0 with b 2 − 4ac ≥ 0
a a
Similarly, both the roots are negative i.e. they lie in (–∞, 0), iff roots are real, the sum of the roots is negative and the product of the roots is positive.
−b c
i.e., α + β= < 0 and αβ= > 0 with b 2 − 4ac ≥ 0
a a
2. Both the roots are greater than a given number k, iff the following conditions are satisfied
−b
D ≥ 0, > k and af (k ) > 0
2a

–b/2a
X′ X X′ X
k (–b/2a) k
3. Both the roots are less than a given number k, iff the following conditions are satisfied
−b
D ≥ 0, < k and af (k ) > 0
2a
09
4. Both the roots lie in a given interval (k1, k2), iff the following conditions are satisfied
−b
D ≥ 0, k1 < < k2 and af ( k1 ) > 0, af ( k2 ) > 0
2a
or f (k1) · f (k2) > 0

X′ X (–b/2a)
X′ X
k1 (–b/2a) k2 k1 k2
5. Exactly one of the roots lie in a given interval (k1, k2), iff D > 0 and f (k1) f (k2) < 0.

k2 k1
X′ X X′ X
k1 k2

6. A given number k lies between the roots, iff af (k) < 0 and D > 0.

k
X′ X
 Questions

1 2 5
1. If is a root of the equation x + kx − =0 , then the value of k is 6. If p and q are the roots of the equation x2 – px + q = 0, then what are
2 4
the value of p and q respectvely?
(a) 2 (b) –2
(a) 1, 0 (b) 0, 1
1 1 (c) –2, 0 (d) –2, 1
(c) (d)
4 2
2. Which of the following equation has the sum of its roots as 3? 7. If the equations x2 + kx + 64 = 0 and x2 – 8x + k = 0 have real roots, then
(a) 2x2 – 3x + 6 = 0 (b) –x2 + 3x – 3 = 0 what is the value of k ?
(a) 4 (b) 8
3
(c) 2 x2 − x +1 =0 (d) 3x2 – 3x + 3 = 0 (c) 12 (d) 16
2
3. Values of k for which the quadratic equation 2x2 – kx + k = 0 has equal 8. (x2 + 1)2 – x = 0 has
roots is (a) four real roots (b) two real roots
(a) 0 only (b) 4 (c) no real roots (d) one real root
(c) 8 only (d) 0, 8
9. If a and b are the roots of the equation x2 + x + 1 = 0, then which of the
4. If the roots of ax2 + bx + c = 0 are sin a and cos a for some a, then following are the roots of the equation x2 – x + 1 = 0?
which one of the following is correct? (a) a7 and b13 (b) a13 and b7
(a) a2 + b2 = 2ac (b) b2 – c2 = 2ab (c) a20 and b20 (d) None of these
(c) b – a = 2ac
2 2
(d) b + c = 2ab
2 2

10. What is the solution set for the equation x4 – 26x2 + 25 = 0

5. If the equation x2 – bx + 1 = 0 does not possess real roots, then which (a) {–5, –1, 1, 5} (b) {–5, –1}
one of the following is correct? (c) {1, 5} (d) {–5, 0, 1, 5}
(a) –3 < b < 3 (b) –2 < b < 2
(c) b > 2 (d) b < – 2
 Answer Key

1. (a)

2. (b)

3. (d)

4. (c)

5. (b)

6. (a)

7. (d)

8. (c)

9. (d)

10. (a)