This document discusses the nature of roots of quadratic equations based on the discriminant. It defines the discriminant as the number used to describe the nature of roots, with the formula d = b^2 - 4ac. It then outlines the nature of roots for different values of the discriminant: if d > 0 the roots are two real and unequal, if d = 0 the root is one real, and if d < 0 there are no real roots. It provides examples of finding the discriminant and describing the nature of roots for quadratic equations.

1. QUARTER 1 MODULE 2
LESSON 1
NATURE OF ROOTS OF
QUADRATIC EQUATION

2. Nature of Roots of Quadratic Equation
Discriminant
• It is the number being
used to describe the
nature of roots of a
quadratic equation.
Formula:
d = b2 – 4ac
provided that the
equation is in standard
form.
discriminant Nature of Roots
d > 0
d is a perfect
square
Two real, rational and
unequal roots
d is not a
perfect
square
Two real, irrational and
unequal roots
d = 0 One real and rational root.
d < 0 No real root

3. discriminant Nature of Roots
d > 0
d is a
perfect
square
• Two real, rational and
unequal roots
d is not a
perfect
square
• Two real, irrational and
unequal roots
d = 0 • One real and rational root.
d < 0 • No real root
Remember: Equation must be in standard form Example 1: Describe the nature of roots
of 3t2 + 5t = 2
Solution:
3t2 + 5t = 2
3t2 + 5t – 2 = 0
a = 3, b = 5, c = –2
d = b2 – 4ac
d = (5)2 – 4(3)(–2)
d = 25 + 24
d = 49
Nature of Roots:
Two real, rational and unequal roots

4. Remember: Equation must be in standard form Example 2: Compute for the discriminant
of 9x2 – 12x + 4 = 0.
Solution:
9x2 – 12x + 4 = 0
a = 9, b = –12, c = 4
d = b2 – 4ac
d = (–12)2 – 4(9)(4)
d = 144 – 144
d = 0
Nature of Roots:
One real and rational root.
discriminant Nature of Roots
d > 0
d is a
perfect
square
• Two real, rational and
unequal roots
d is not a
perfect
square
• Two real, irrational and
unequal roots
d = 0 • One real and rational root.
d < 0 • No real root

5. Remember: Equation must be in standard form Example 3: What is the nature of roots of
h2 = 16 – 3h
Solution:
h2 = 16 – 3h
h2 + 3h – 16 = 0
a = 1, b = 3, c = –16
d = b2 – 4ac
d = (3)2 – 4(1)(–16)
d = 9 + 64
d = 73
Nature of Roots:
Two real, irrational and unequal roots
discriminant Nature of Roots
d > 0
d is a
perfect
square
• Two real, rational and
unequal roots
d is not a
perfect
square
• Two real, irrational and
unequal roots
d = 0 • One real and rational root.
d < 0 • No real root

6. Remember: Equation must be in standard form Example 4: Describe the nature of roots
of the equation 4k2 + 6k + 3 = 0
Solution:
4k2 + 6k + 3 = 0
a = 4, b = 6, c = 3
d = b2 – 4ac
d = (6)2 – 4(4)(3)
d = 36 – 48
d = –12
Nature of Roots:
No real root
discriminant Nature of Roots
d > 0
d is a
perfect
square
• Two real, rational and
unequal roots
d is not a
perfect
square
• Two real, irrational and
unequal roots
d = 0 • One real and rational root.
d < 0 • No real root

7. Remember: Equation must be in standard form Example 5: Determine the nature of roots
of the equation 3x2 + 11x + 8 = 0
Solution:
3x2 + 11x + 8 = 0
a = 3, b = 11, c = 8
d = b2 – 4ac
d = (11)2 – 4(3)(8)
d = 121 – 96
d = 25
Nature of Roots:
Two real, rational and unequal roots
discriminant Nature of Roots
d > 0
d is a
perfect
square
• Two real, rational and
unequal roots
d is not a
perfect
square
• Two real, irrational and
unequal roots
d = 0 • One real and rational root.
d < 0 • No real root

8. Answer Activity 1.3: What’s More? on
page 6 of your Quarter 1 Module 2

9. Thank you.