Illustrating Quadratic Equations

2. What is a QUADRATIC EQUATION?
a second-degree equation in one variable that can be
written in the form 𝒂𝒙𝟐
+ 𝒃𝒙 + 𝒄 = 𝟎. Where 𝑎, 𝑏, and 𝑐
are real numbers and 𝑎 ≠ 0.
𝒂𝒙𝟐
is the quadratic term
𝒃𝒙 is the linear term
𝒄 is the constant term

3. How to identify QUADRATIC EQUATIONS?
It must be on the second degree or the highest exponent
of the variable must be two (2).
It must involve one variable only.
No variable in the
denominator.
No variable in the exponent.
No variable inside a radical sign.

4. Examples 1.𝒙𝟐
+ 𝟑𝒙 + 𝟔 = 𝟎
This is a quadratic equation since the highest
exponent is 2.
It is written in standard form (𝑎𝑥2
+ 𝑏𝑥 + 𝑐 = 0)
The values of 𝑎, 𝑏, and 𝑐 are 𝑎 = 1, 𝑏 = 3, 𝑐 = 6

5. Examples 1. 𝒙𝟐
+ 𝟑𝒙 + 𝟔 = 𝟎
This is a quadratic equation since the highest exponent is 2.
It is written in standard form (𝑎𝑥2
+ 𝑏𝑥 + 𝑐 = 0)
The values of 𝑎, 𝑏, and 𝑐 are 𝑎 = 1, 𝑏 = 3, 𝑐 = 6
2. 𝟒𝒙𝟐
− 𝟓𝒙 = 𝟖
This is a quadratic equation since the highest degree of the
equation is 2.
The given is not written in standard form. The standard form of
the given equation is
4𝑥2
− 5𝑥 − 8 = 0
The values of 𝑎, 𝑏, and 𝑐 are 𝑎 = 4, 𝑏 = −5, 𝑐 = −8
𝟒𝒙𝟐
− 𝟓𝐱 = 𝟖
𝟒𝒙𝟐
− 𝟓𝐱 − 𝟖 = 𝟖 − 𝟖
𝟒𝒙𝟐
− 𝟓𝐱 − 𝟖 = 𝟎

6. Examples 1.𝒙𝟐
+ 𝟑𝒙 + 𝟔 = 𝟎
This is a quadratic equation since the highest exponent is 2.
It is written in standard form (𝑎𝑥2
+ 𝑏𝑥 + 𝑐 = 0)
The values of 𝑎, 𝑏, and 𝑐 are 𝑎 = 1, 𝑏 = 3, 𝑐 = 6
2.𝟒𝒙𝟐
− 𝟓𝒙 = 𝟖
This is a quadratic equation since the highest degree of the
equation is 2.
The given is not written in standard form. The standard form of
the given equation is
4𝑥2
− 5𝑥 − 8 = 0
The values of 𝑎, 𝑏, and 𝑐 are 𝑎 = 4, 𝑏 = −5, 𝑐 = −8
3. 𝟗𝒙𝟐
− 𝟏𝟗 = 𝟎
This is a quadratic equation written in standard form.
The values of 𝑎, 𝑏, and 𝑐 are 𝑎 = 9, 𝑏 = 0, 𝑐 = −19

7. Examples
4. 𝟑𝒙(𝒙 + 𝟓) = 𝟎 This is a quadratic equation but not written in standard form.
The standard form of the given equation is
3𝑥2
+ 15𝑥 = 0
The values of 𝑎, 𝑏, and 𝑐 are 𝑎 = 3, 𝑏 = 15, 𝑐 = 0
𝟑𝒙 𝒙 + 𝟓 = 𝟎
𝟑𝒙 𝒙) + 𝟑𝒙(𝟓 = 𝟎
𝟑𝒙𝟐
+ 𝟏𝟓𝒙 = 𝟎

8. Examples
4. 𝟑𝒙(𝒙 + 𝟓) = 𝟎 This is a quadratic equation but not written in standard form.
The standard form of the given equation is
3𝑥2
+ 15𝑥 = 0
The values of 𝑎, 𝑏, and 𝑐 are 𝑎 = 3, 𝑏 = 15, 𝑐 = 0
5. 𝟏𝟏𝒙−𝟐
+ 𝟑𝒙 + 𝟕 = 𝟎 This equation is not a quadratic equation since there is a
negative exponent.

9. Examples
4. 𝟑𝒙(𝒙 + 𝟓) = 𝟎 This is a quadratic equation but not written in standard form.
The standard form of the given equation is
3𝑥2
+ 15𝑥 = 0
The values of 𝑎, 𝑏, and 𝑐 are 𝑎 = 3, 𝑏 = 15, 𝑐 = 0
5. 𝟏𝟏𝒙−𝟐
+ 𝟑𝒙 + 𝟕 = 𝟎 This equation is not a quadratic equation since there is a
negative exponent.
6. 𝟑𝒙 + 𝟔 = 𝟎 This is not a quadratic equation but a linear equation

10. Activity 1 Organize the equations
From the pool below, choose the quadratic equations. Explain why you
pick that equation as a quadratic equation.
𝒙𝟐 + 𝟓𝒙 + 𝟓
𝒙 + 𝒚 = 𝟐
𝒙𝟐 + 𝟗𝒙 + 𝟏𝟏 = 𝟎
𝒙 + 𝟏 = 𝟎 𝟑𝒙𝟐 − 𝟐𝟏𝟏 = 𝟒𝒙 𝒙𝟐
+ 𝟗𝒚 + 𝟏𝟏 = 𝟎
𝒙𝟐 = 𝟐 𝒙𝟑
− 𝟐𝒙𝟐
+ 𝟒𝒙 − 𝟑 = 𝟎 𝒙𝟐
+ 𝟐𝒙 + 𝟑 = 𝟎
𝟎 = 𝒙𝟐
𝒚−𝟐
+ 𝟓𝒚 + 𝟑 = 𝟎 𝟔𝒙𝟐 + 𝟏𝟐𝒙 + 𝟐𝟒 = 𝟎
𝒙𝟐+𝟏𝟓 = 𝟎
𝟒𝒙𝟐 − 𝟓 = 𝟐𝟐
𝟏𝟐𝟐
= 𝟐𝟐𝒙 + 𝒙𝟐
𝒚
𝟏
𝟐 + 𝟓𝒚 = 𝟏𝟔 𝒙𝟐
+ 𝟐𝒙 = 𝟎

11. Activity 2 It’s time to sort things out.
Change the given quadratic equations in standard form, identify the
values of a, b, and c.
Quadratic Equation
Standard Form
(𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄 = 𝟎)
a b c
1. 𝒙𝟐 + 𝟗𝒙 + 𝟏𝟏 = 𝟎

12. Activity 2 It’s time to sort things out.
Change the given quadratic equations in standard form, identify the
values of a, b, and c.
Quadratic Equation
Standard Form
(𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄 = 𝟎)
a b c
1. 𝒙𝟐 + 𝟗𝒙 + 𝟏𝟏 = 𝟎 𝒙𝟐 + 𝟗𝒙 + 𝟏𝟏 = 𝟎 1 9 11

13. Activity 2 It’s time to sort things out.
Change the given quadratic equations in standard form, identify the
values of a, b, and c.
Quadratic Equation
Standard Form
(𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄 = 𝟎)
a b c
1. 𝒙𝟐 + 𝟗𝒙 + 𝟏𝟏 = 𝟎 𝒙𝟐 + 𝟗𝒙 + 𝟏𝟏 = 𝟎 1 9 11
𝟐. 𝟑𝒙𝟐
− 𝟐𝟏𝟏 = 𝟒𝒙
𝟑𝒙𝟐
− 𝟐𝟏𝟏 = 𝟒𝒙
𝟑𝒙𝟐
− 𝟐𝟏𝟏 − 𝟒𝒙 = 𝟒𝒙 −𝟒𝒙
𝟑𝒙𝟐
− 𝟐𝟏𝟏 − 𝟒𝒙 = 𝟎
𝟑𝒙𝟐
− 𝟒𝒙 − 𝟐𝟏𝟏 = 𝟎

14. Activity 2 It’s time to sort things out.
Change the given quadratic equations in standard form, identify the
values of a, b, and c.
Quadratic Equation
Standard Form
(𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄 = 𝟎)
a b c
1. 𝒙𝟐 + 𝟗𝒙 + 𝟏𝟏 = 𝟎 𝒙𝟐 + 𝟗𝒙 + 𝟏𝟏 = 𝟎 1 9 11
𝟐. 𝟑𝒙𝟐
− 𝟐𝟏𝟏 = 𝟒𝒙 𝟑𝒙𝟐 − 𝟒𝒙 − 𝟐𝟏𝟏 = 𝟎 3 -4 -211
𝟑. 𝟎 = 𝒙𝟐

15. Activity 2 It’s time to sort things out.
Change the given quadratic equations in standard form, identify the
values of a, b, and c.
Quadratic Equation
Standard Form
(𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄 = 𝟎)
a b c
1. 𝒙𝟐 + 𝟗𝒙 + 𝟏𝟏 = 𝟎 𝒙𝟐 + 𝟗𝒙 + 𝟏𝟏 = 𝟎 1 9 11
𝟐. 𝟑𝒙𝟐
− 𝟐𝟏𝟏 = 𝟒𝒙 𝟑𝒙𝟐 − 𝟒𝒙 − 𝟐𝟏𝟏 = 𝟎 3 -4 -211
𝟑. 𝟎 = 𝒙𝟐
𝒙𝟐
= 𝟎 1 0 0

16. Activity 2 It’s time to sort things out.
Change the given quadratic equations in standard form, identify the
values of a, b, and c.
Quadratic Equation
Standard Form
(𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄 = 𝟎)
a b c
1. 𝒙𝟐 + 𝟗𝒙 + 𝟏𝟏 = 𝟎 𝒙𝟐 + 𝟗𝒙 + 𝟏𝟏 = 𝟎 1 9 11
𝟐. 𝟑𝒙𝟐
− 𝟐𝟏𝟏 = 𝟒𝒙 𝟑𝒙𝟐 − 𝟒𝒙 − 𝟐𝟏𝟏 = 𝟎 3 -4 -211
𝟑. 𝟎 = 𝒙𝟐
𝒙𝟐
= 𝟎 1 0 0
4.𝒙𝟐
+ 𝟏𝟓 = 𝟎

17. Activity 2 It’s time to sort things out.
Change the given quadratic equations in standard form, identify the
values of a, b, and c.
Quadratic Equation
Standard Form
(𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄 = 𝟎)
a b c
1. 𝒙𝟐 + 𝟗𝒙 + 𝟏𝟏 = 𝟎 𝒙𝟐 + 𝟗𝒙 + 𝟏𝟏 = 𝟎 1 9 11
𝟐. 𝟑𝒙𝟐
− 𝟐𝟏𝟏 = 𝟒𝒙 𝟑𝒙𝟐 − 𝟒𝒙 − 𝟐𝟏𝟏 = 𝟎 3 -4 -211
𝟑. 𝟎 = 𝒙𝟐
𝒙𝟐
= 𝟎 1 0 0
4.𝒙𝟐
+ 𝟏𝟓 = 𝟎 𝒙𝟐
+ 𝟏𝟓 = 𝟎 1 0 15

18. Activity 2 It’s time to sort things out.
Change the given quadratic equations in standard form, identify the
values of a, b, and c.
Quadratic Equation
Standard Form
(𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄 = 𝟎)
a b c
1. 𝒙𝟐 + 𝟗𝒙 + 𝟏𝟏 = 𝟎 𝒙𝟐 + 𝟗𝒙 + 𝟏𝟏 = 𝟎 1 9 11
𝟐. 𝟑𝒙𝟐
− 𝟐𝟏𝟏 = 𝟒𝒙 𝟑𝒙𝟐 − 𝟒𝒙 − 𝟐𝟏𝟏 = 𝟎 3 -4 -211
𝟑. 𝟎 = 𝒙𝟐
𝒙𝟐
= 𝟎 1 0 0
4.𝒙𝟐
+ 𝟏𝟓 = 𝟎 𝒙𝟐
+ 𝟏𝟓 = 𝟎 1 0 15
𝟓. 𝒙𝟐
+ 𝟐𝒙 = 𝟎

19. Activity 2 It’s time to sort things out.
Change the given quadratic equations in standard form, identify the
values of a, b, and c.
Quadratic Equation
Standard Form
(𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄 = 𝟎)
a b c
1. 𝒙𝟐 + 𝟗𝒙 + 𝟏𝟏 = 𝟎 𝒙𝟐 + 𝟗𝒙 + 𝟏𝟏 = 𝟎 1 9 11
𝟐. 𝟑𝒙𝟐
− 𝟐𝟏𝟏 = 𝟒𝒙 𝟑𝒙𝟐 − 𝟒𝒙 − 𝟐𝟏𝟏 = 𝟎 3 -4 -211
𝟑. 𝟎 = 𝒙𝟐
𝒙𝟐
= 𝟎 1 0 0
4.𝒙𝟐
+ 𝟏𝟓 = 𝟎 𝒙𝟐
+ 𝟏𝟓 = 𝟎 1 0 15
𝟓. 𝒙𝟐
+ 𝟐𝒙 = 𝟎 𝒙𝟐 + 𝟐𝒙 = 𝟎 1 2 0