1. YES-NO-STAND-UP Game
Questions:
1. Do you like chocolates?
2. Do you like to hurt
people?
3. Do you love math?
4. Is 2 + 2 = 5?
5. Is 9= 3 ?
6. Is 4 a perfect square?
7. Is 1 x 2 = 3?
8. Is 100 not a perfect
square?
9. Is there a way to predict
the nature of the root?
10. Are you familiar with
discriminant in a quadratic?
equation?
3. At the end of this lesson, the students are expected to:
Objectives
b. describes the nature of roots of a quadratic
equation through its discriminant;
c. value the importance of discriminant through the
roots of the quadratic equation.
a. identifies the nature of roots of a given quadratic
equation;
4. Group Activity
1. 𝒙𝟐
+ 𝟒𝒙 + 𝟑 = 𝟎 2. 𝒙𝟐
− 𝟏 = 𝟎
Boys vs. Girls
Direction: Find the roots of the quadratic eaquation
using quadratic formula. Boys group will answer #1 and
girls group will answer the #2. The groups have 5
minutes to answer and choose a representative to
present their answer.
5. Questions from the activity:
1. What is the standard form of the quadratic equation?
2. What is the formula for quadratic formula?
3. What is the formula of a discriminant?
4. Why do we need to determine the discriminant
of the quadratic equation?
6.
7. WHY USE THE QUADRATIC FORMULA?
The quadratic formula allows you to solve ANY quadratic
equation, even if you cannot factor it.
An important piece of the quadratic formula is what’s under the radical:
b2 – 4ac
This piece is called the discriminant.
8. WHY IS THE DISCRIMINANT IMPORTANT?
The value of the expression 𝒃𝟐
– 4ac is
called the discriminant of the quadratic
equation ax² + bx + c = 0. This value can be
used to describe the nature of the roots
of a quadratic equation. It can be zero,
positive and perfect square, positive but
not perfect square or negative.
???
9. 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
real numbers and equal
roots
d > 0
d is a perfect
square
rational numbers and
unequal roots
d is not a
perfect
square
irrational numbers and
unequal roots
d < 0 No real roots
10. 9
Remember: Equation must be in standard form Example 1: Describe the nature of roots
of 3t2 + 5t = 2
Solution:
3t2 + 5t = 2
Step 1: Write first the equation into
standard form: 3t2 + 5t – 2 = 0
Step 2: Identify a, b and c
a = 3, b = 5, c = –2
Step 3: Substitute these values to
d = b2 – 4ac
d = (5)2 – 4(3)(–2)
d = 25 + 24
d = 49
Nature of Roots:
rational numbers and equal
discriminant Nature of Roots
d = 0
real numbers and equal
roots
d > 0
d is a perfect
square
rational and unequal roots
d is not a
perfect
square
irrational and unequal roots
d < 0 No real root
11. Remember: Equation must be in standard form Example 2: Compute for the discriminant
of (x – 2)2 = 0
Solution:
Step 1: Write first the equation into
standard form: x2 – 4x + 4 = 0
Step 2: Identify a, b and c
a = 1, b = –4, c = 4
Step 3: Substitute these values to
d = b2 – 4ac
d = (–4)2 – 4(1)(4)
d = 16 – 16
d = 0
Nature of Roots:
real numbers and equal
discriminant Nature of Roots
d = 0 real numbers and equal
d > 0
d is a
perfect
square
rational numbers and
unequal
d is not a
perfect
square
irrational numbers and
unequal
d < 0 No real roots
12. Remember: Equation must be in standard form Example 3: What is the nature of roots of
h2 – 16 = – 3h
Solution:
Step 1: Write first the equation into
standard form: h2 + 3h – 16 = 0
Step 2: Identify a, b and c
a = 1, b = 3, c = –16
Step 3: Substitute these values to
d = b2 – 4ac
d = (3)2 – 4(1)(–16)
d = 9 + 64
d = 73
Nature of Roots:
irrational numbers and unequal
discriminant Nature of Roots
d = 0 real numbers and equal
d > 0
d is a
perfect
square
rational numbers and
unequal
d is not a
perfect
square
irrational numbers and
unequal
d < 0 No real roots
13. Remember: Equation must be in standard form Example 4: Describe the nature of roots
of the equation k2 - 3k = - 6
Solution:
Step 1: Write first the equation into
standard form: k2 - 3k + 6 = 0
Step 2: Identify a, b and c
a = 1, b = -3, c = 6
Step 3: Substitute these values to
d = b2 – 4ac
d = (-3)2 – 4(1)(6)
d = 9 – 24
d = – 15
Nature of Roots:
No real 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
discriminant Nature of Roots
d = 0 real numbers and equal
d > 0
d is a
perfect
square
rational numbers and
unequal roots
d is not a
perfect
square
irrational numbers and
unequal roots
d < 0 No real root
14. Remember: Equation must be in standard form Example 5: Determine the nature of roots
of the equation 3x2 + 11x + 8 = 0
Solution:
Step 1: Write first the equation into
standard form: 3x2 + 11x + 8 = 0
Step 2: Identify a, b and c
a = 3, b = 11, c = 8
Step 3: Substitute these values to
d = b2 – 4ac
d = (11)2 – 4(3)(8)
d = 121 – 96
d = 25
Nature of Roots:
rational numbers and unequal
discriminant Nature of Roots
d = 0 real numbers and equal
d > 0
d is a
perfect
square
rational numbers and
unequal roots
d is not a
perfect
square
irrational and unequal
roots
d < 0 No real root
15. Activity
Place Me on the Table!
Direction: Answer the following.
1. Complete the table below.
Equation b²-4ac Nature of roots
1. x² + 5x + 4 = 0
2. 2x² + x - 21= 0
3. 4x² +4x + 1 = 0
4. x² - 2x - 2 = 0
5. 9x² + 16= 0
2. How would you describe the roots of
quadratic equation when the value of b²-
4ac is 0? Positive and Perfect Square?
Positive but not Perfect Square? Negative?
.
3. Which quadratic equation has
roots that are real numbers and
equal? Rational numbers? Irrational
numbers? Not real numbers?
4. How do you determine the
quadratic equation having roots that
are real numbers and equal?
Rational numbers? Irrational
numbers? Not real numbers?
16. Directions: Determine the discriminants and nature of the roots of the following
quadratic equations. Write your answer on the space provided.
Evaluation
2. 𝑥2
+ 9𝑥 + 20 = 0
1. 𝑥2 + 6𝑥 + 9 = 0
3. 2𝑥2 − 10𝑥 + 8 = 0
4. 𝑥2 + 5𝑥 + 10 = 0
5. 𝑥2
+ 6𝑥 + 3 = 0
discriminant:
discriminant:
discriminant:
discriminant:
discriminant: nature of the roots
nature of the roots
nature of the roots
nature of the roots
nature of the roots
