This document discusses using the discriminant of a quadratic equation (b^2 - 4ac) to characterize the nature of its roots. It defines the discriminant and explains that if the discriminant is equal to 0, the roots are real and equal. If the discriminant is positive and a perfect square, the roots are rational and unequal, and if positive but not a perfect square the roots are irrational and unequal. If the discriminant is negative, there are no real roots. Examples are provided to illustrate these concepts.

1. NATURE OF THE ROOTS
OF A QUADRATIC
EQUATION USING THE
DISCRIMINANT

2. LESSON OBJECTIVE
Characterize the roots of a
quadratic equation using
discriminant.

3. SHORT ACTIVITY
Write the following quadratic equations in standard form
ax2 + bx + c = 0 , and identify the value of a, b, and c.
1. x2 − 9x = 16

4. SHORT ACTIVITY
Write the following quadratic equations in standard form
ax2 + bx + c = 0 , and identify the value of a, b, and c.
2. 3 x2 + 3x − 5 = 0

5. SHORT ACTIVITY
Write the following quadratic equations in standard form
ax2 + bx + c = 0 , and identify the value of a, b, and c.
3. 5x + 2 2x − 5 = 𝟎

6. DISCRIMINANT
The value of the expression b
2
− 4ac is
called the discriminant of a quadratic
equation ax2 + bx + c = 0. This value
can be used to describe the nature of
the roots of a quadratic equation.

7. NATURE OF THE ROOTS OF A
QUADRATIC EQUATION
If the value of b
2
− 4ac is
equal to zero, then the roots
are real and equal.

8. NATURE OF THE ROOTS OF A
QUADRATIC EQUATION
If the value of b
2
− 4ac is
positive and perfect square,
then the roots are rational
number and unequal.

9. NATURE OF THE ROOTS OF A
QUADRATIC EQUATION
If the value of b
2
− 4ac is
positive but not perfect square,
then the roots are irrational
and unequal.

10. NATURE OF THE ROOTS OF A
QUADRATIC EQUATION
If the value of b
2
− 4ac is
negative, then it has no real
roots.

11. EXAMPLE #1
x2 + 12x + 36 = 0

12. EXAMPLE #2
4x2 − 20x + 25 = 0

13. EXAMPLE #3
x2 − 2x − 48 = 0

14. EXAMPLE #4
3x2 + 5x − 12 = 0

15. EXAMPLE #5
x2 + 7x + 11 = 0

16. ACTIVITY: LOOK THEN FILL ME UP
Consider the given equations and fill in
the table below.
Quadratic Equation Quadratic Equation in
Standard Form
Value of 𝒃𝟐
−
𝟒𝒂𝒄
Nature of the
Roots
1. 𝒙𝟐 + 𝟑𝒙 + 𝟏𝟐 = 𝟎
2. 𝒙𝟐 − 𝟔𝒙 + 𝟗 = 𝟎
3. 𝟑𝒙𝟐 = −𝟐𝒙 − 𝟓
4. 𝟑𝒙𝟐 − 𝟏𝟐 = 𝟕𝒙
5. 𝟐𝒙𝟐 − 𝟔𝒙 = −𝟏𝟐

17. EXAMPLE #6
2x2 − 17x + 18 = 0

18. EXAMPLE #7
x2 + 4x + 15 = 0

19. EXAMPLE #8
3x2 − 5x + 9 = 0

20. UP NEXT:
Describe the relationship
between the coefficients and
the roots of a quadratic
equation.

22. ACTIVITY: LOOK THEN FILL ME UP
Consider the given equations and fill in
the table below.
Quadratic Equation Quadratic Equation in
Standard Form
Value of 𝒃𝟐
−
𝟒𝒂𝒄
Nature of the
Roots
1. 𝒙𝟐 + 𝟑𝒙 + 𝟏𝟐 = 𝟎
2. 𝒙𝟐 − 𝟔𝒙 + 𝟗 = 𝟎
3. 𝟑𝒙𝟐 = −𝟐𝒙 − 𝟓
4. 𝟑𝒙𝟐 − 𝟏𝟐 = 𝟕𝒙
5. 𝟐𝒙𝟐 − 𝟔𝒙 = −𝟏𝟐