1. Nature of Roots of a Quadratic Equation

2. First of all, let’s recall the quadratic formula.
Yes, in fact, the nature of roots can be determined by simply the value of an expression.
Do you remember what nature of roots of a quadratic equation is?
Does it refer to whether the roots are real or not real, equal or unequal?

3. Discriminant of a Quadratic Equation
Consider a quadratic equation:
ax2 + bx + c = 0, where a  0
The value of this expression can determine the nature of roots. a 2 ac 4 b x - =
Its roots are given by:
 Quadratic formula
The expression is called the discriminant
(denoted by ).
 pronounced as ‘delta’
Discriminant:  = b2  4ac

4. ac b 4 - Case 1:  > 0  i.e. b2 – 4ac > 0 ∵
is a positive real number. ∵ ∴
The roots of the quadratic equation are
and . a ac b 2 4 - + 
The two roots are
real and unequal.
two unequal real roots

5. ac b 4 - Case 1:  > 0  i.e. b2 – 4ac > 0 ∵
is a positive real number. ∵ ∴
The roots of the quadratic equation are
and . a ac b 2 4 - + 
∴ The equation has two unequal real roots.
e.g.
Consider x2 + 3x – 7 = 0.
= 32 – 4(1)(–7)
= 37
> 0
∴ The equation x2 + 3x – 7 = 0 has two unequal real roots.

6. ac b 4 - Case 2:  = 0  i.e. b2 – 4ac = 0 ∵ is zero. ∴
The roots of the quadratic equation are . a b 2 - . a b 2  - ac b 4 2 -
The two roots are
real and equal.
one double real root

7. ac b 4 - Case 2:  = 0  i.e. b2 – 4ac = 0 ∵ is zero. ∴
The roots of the quadratic equation are . a b 2 -
∴ The equation has one double real root.
e.g.
Consider x2 – 8x + 16 = 0.
= (–8)2 – 4(1)(16)
= 0
∴ The equation x2 – 8x + 16 = 0 has one double real root.

8. ac b 4 - Case 3:  < 0  i.e. b2 – 4ac < 0 is not a real number.∵ ∴
The roots of the quadratic equation are
not real.
∴ The equation has no real roots.
The two roots are
no real roots.

9. ac b 4 - Case 3:  < 0  i.e. b2 – 4ac < 0 is not a real number.∵ ∴
The roots of the quadratic equation are
not real.
∴ The equation has no real roots.
e.g.
Consider x2 – 2x + 5 = 0.
= (–2)2 – 4(1)(5)
= –16
< 0
∴ The equation x2 – 2x + 5 = 0 has no real roots.

10. The table below summarizes the three cases of the nature of roots of a quadratic equation.
Condition
Nature of its roots
 > 0
 = 0
 < 0
two unequal
real roots
one double
real root
no real roots
(or two distinct
real roots)
(or two equal
real roots)

11. Follow-up question
Find the value of the discriminant of the equation x2 – 5x + 3 = 0, and hence determine the nature of its roots.
The discriminant of x2 – 5x + 3 = 0 is given by: 13 ) 3 )( 1 ( 4 5 2 = - D
>
∴ The equation has two distinct real roots.

12. Graph of a Quadratic Equation
For the quadratic equation ax2 + bx + c = 0, are there any relations among the following:
discriminant,
nature of roots,
no. of x-intercepts of the corresponding graph.

13. For the quadratic equation ax2 + bx + c = 0:
2 unequal real roots
1 double real root
0 real roots
Value of the discriminant
determine
nature of roots of ax2 + bx + c = 0
no. of x-intercepts of the graph of y = ax2 + bx + c
2 x-intercepts
1 x-intercept
0 x-intercepts
Roots of ax2 + bx + c = 0
equals
no. of x-intercepts of the graph of y = ax2 + bx + c
value of the discriminant
Therefore
also tells us

14. No. of x-intercepts of the graph of y = ax 2 + bx + c
The discriminant of the quadratic equation ax2 + bx + c = 0, the nature of its roots and the number of x-intercepts of the graph of y = ax2 + bx + c have the following relations.
Discriminant
( = b 2  4ac)
Nature of roots of
ax 2 + bx + c = 0
No. of x-intercepts of the graph of y = ax 2 + bx + c
2 unequal real roots
1 double real root
no real roots 2 1
 > 0
 = 0
 < 0

15. Follow-up question
Some graphs of y = ax2 + bx + c will be shown one by one. For each corresponding quadratic equation ax2 + bx + c = 0, determine whether  > 0,  = 0 or  < 0. O y x O y x
 > 0
2 x-intercepts
 > 0
2 x-intercepts O y x O x y
1 x-intercept
 < 0
 = 0
no x-intercepts