Quadratic Equations

3. QUADRATIC
EQUATION
Mark Joven A. Alam-alam, LPT
In this lesson, you will learn to:
• illustrate quadratic equations;
• convert from standard form of a quadratic equation to its
general form and vice versa;
• characterize the roots of a quadratic equation using the
discriminant;
• describe the relationship between the coefficients and the
roots of a quadratic equation.

4. QUADRATIC EQUATION
•an equation that can be written in the form
𝒂𝒙𝟐
+ bx + c = 0, where a, b, and c are
real numbers, and a is not equal to 0
Standard form of a Quadratic Equation
𝒂𝒙𝟐
+ bx + c = 0

5. QUADRATIC EQUATION
•if in one variable is known as an equation of
degree 2
𝒂𝒙𝟐
+ bx + c = 0
quadratic term
linear term
constant

6. EXAMPLE 1: Rewrite the equation in standard form to
determine if it is a quadratic equation. Then, identify
the coefficients of the quadratic and linear terms
and the constant.
1. 24x - 𝑥2
= 9
2. (𝑥 + 1)3
= 0
3.
1
𝑥2+4𝑥+4
= 0
4. 1 – 2x = 3𝑥2

7. TRY THIS: Rewrite the equation in standard form to
determine if it is a quadratic equation. Then, identify
the coefficients of the quadratic and linear terms
and the constant.
1. 6𝑥2- 3x + 9 = 0
2. 12𝑥3
− 1 = 3x(4x + 2)

9. Roots of the Quadratic Equation
•all the values of the variable that
satisfy a given quadratic equation
in one variable
•solution of the equation
•maximum of two roots

10. •used to determine the
number and nature of the
roots of a quadratic
equation
discriminant 𝑏2
- 4ac

11. Number & Nature of the Roots
Given a quadratic equation 𝒂𝒙𝟐
+ bx + c = 0, where a ≠ 0:
If 𝑏2
- 4ac = 0
•The roots are equal and are both rational numbers
•There is only one real and rational solution.

12. Number & Nature of the Roots
Given a quadratic equation 𝒂𝒙𝟐
+ bx + c = 0, where a ≠ 0:
If 𝑏2
- 4ac > 0 and is a perfect square
•There are two distinct real roots that are both rational
numbers.

13. Number & Nature of the Roots
Given a quadratic equation 𝒂𝒙𝟐
+ bx + c = 0, where a ≠ 0:
If 𝑏2
- 4ac > 0 but is not a perfect square
•There are two distinct real roots that are both irrational
numbers.

14. Number & Nature of the Roots
Given a quadratic equation 𝒂𝒙𝟐
+ bx + c = 0, where a ≠ 0:
If 𝑏2
- 4ac < 0
•Has no real roots
•Roots are two distinct imaginary numbers

15. EXAMPLE 2: Determine the number and nature of
roots of the following quadratic equations by
solving for their discriminants.
1. 4𝑥2
+ 8x + 9 = 0
2. −𝑚2
+ 2m – 11 = -10
3. 6𝑝2 = 4 – 2p
4. 3𝑝2 - p – 3 = 0

16. TRY THIS: Rewrite the equation in standard form to
determine if it is a quadratic equation. Then, identify
the coefficients of the quadratic and linear terms
and the constant.
1. 𝑛2
- 9n + 10 = 0
2. 4𝑥2
+ 4x + 1 = 0
TRY THIS: Determine the number and nature of roots
of the following quadratic equations by solving for
their discriminants.

17. Sum and Products of the Roots
Let 𝑟1 and 𝑟2 be the roots of the quadratic
equation 𝑎𝑥2 + bx + c = 0:
𝒓𝟏 + 𝒓𝟐 = -
𝒃
𝒂
𝒓𝟏𝒓𝟐 =
𝒄
𝒂
Sum of the roots Product of the roots

18. EXAMPLE 3: Determine the sum and products of the
roots of the following quadratic equations.
1. 𝑥2
+ 1 = 2x
2. 3𝑥2
+ 7x + 4 = 0

19. The quadratic equation can also be written in
standard form using the sum and product of its
roots using the following formula:
𝑥2
- (sum of roots)x + (product of roots) = 0

20. EXAMPLE 4: Write the quadratic equation in standard
form given the sum and the product of its roots.
1. sum = 3
product = -5 2. sum = -4
product =
7
3

21. TRY THIS: Rewrite the equation in standard form to
determine if it is a quadratic equation. Then, identify
the coefficients of the quadratic and linear terms
and the constant.
TRY THIS: Write the quadratic equation in standard
form given the sum and the product of its roots.
1. sum = -5
product = 4
2. sum =
3
4
product = -1

22. THINGS TO REMEMBER!!!
•A quadratic equation is an equation
that can be written in the form 𝒂𝒙𝟐
+ bx + c = 0, where a, b, and c
are real numbers, and a is not
equal to 0.

23. THINGS TO REMEMBER!!!
•The number and nature of roots of
a quadratic equation can be
determined by computing for the
discriminant, 𝒃𝟐
- 4ac.

24. THINGS TO REMEMBER!!!
•The coefficients of a
quadratic equation are
related to the sum and
product of its roots.

26. Quadratic Equation
•if in one variable is known as
an equation of degree 2
LOYALTY

27. QUIZ 1
Math Empowerment (Pages 12-13 in your
Math9 book)
Answer the following:
A. Items 1-5
B. Items 1 & 2
C. Items 1 & 2
D. Items 1& 2
Copy and Answer
in your Math9
Activity/Quiz
Notebook

28. EXIT SLIP
Name,
•One thing I learned: ______________
•One thing I still want to know: _______