Quadratic functions are polynomial equations of the form ax^2 + bx + c = 0, where a ≠ 0. They produce U-shaped parabolic curves that open upwards or downwards. Examples are provided of different quadratic equations in standard and disguised forms. Methods are demonstrated for determining if equations are quadratic, expressing equations in standard form, and using the quadratic formula to solve quadratic equations.

1. Quadratic
Functions
𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0

2. Quadratic Functions
Here are examples of different types of parabolas
As you can see looking at the image, all the
parabolas are U-shaped and they open either
upwards or downwards. If you rotate it to open
sideways, this function fails the vertical line test
• Polynomial equation in a single variable where the highest exponent of the variable is 2
• Also known as a parabola (U-shaped and they open in 1 direction)

3. 𝑎𝑥2 − 𝑞𝑢𝑎𝑑𝑟𝑎𝑡𝑖𝑐 𝑡𝑒𝑟𝑚 𝑏𝑥 − 𝑙𝑖𝑛𝑒𝑎𝑟 𝑡𝑒𝑟𝑚 𝑐 − 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡

4. Here are some examples:
a = 2; b = 5; c = 3
a = 1; b = - 3; c = 0
not a quadratic equation

5. In Disguise Quadratic Equations 𝑎𝑥2
+ 𝑏𝑥 + 𝑐 = 0
2𝑥 (4𝑥 + 1) = 3 8𝑥2
+ 2𝑥 − 3 = 0
(𝑥 + 3)2
= 0 𝑥2
+ 6𝑥 + 9 = 0
1
𝑥
=
2𝑥
𝑥 + 4
2𝑥2
− 𝑥 − 4 = 0

6. Determine if the given equations are quadratic
equations.
a. 4𝑥2 + 8𝑥 − 2 = 0
b. 9𝑥2
+ 45 = 0
c. 𝑥3
+ 5𝑥2
− 4 = 0
d. 𝑥 − 4 = 0
e.6𝑥5 + 𝑥3 + 3 = 0
f. 10𝑥2 − 5𝑥 = 12
Quadratic
Quadratic
Non - quadratic
Non - quadratic
Non - quadratic
Quadratic

7. Determine if the given equations are quadratic equations.
a. 𝑥2 + 3𝑥 = 0
b. 2𝑥 − 6 = 0
c. 2 − 5𝑥2 + 𝑥3 = 0
Quadratic
Non - quadratic
Non - quadratic

8. 1. Express the following quadratic equations in the
form of 𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0
a. 𝑥2 = 5𝑥 + 6
b. −𝑥2
= 4𝑥 + 4
c. 3x − 6𝑥2 = 7
d. −𝑥2 − 2 = 3𝑥
𝑥2 − 5𝑥 − 6 = 0
𝑥2
+ 4𝑥 + 4 = 0
6𝑥2 − 3𝑥 + 7 = 0
𝑥2 + 3𝑥 + 2 = 0

9. 2. Show that 5x(x+3) = 4x – 5 is a quadratic equation.
5x(x+3) = 4x – 5
5𝑥2 + 15𝑥 = 4𝑥 − 5
5𝑥2 + 15𝑥 − 4𝑥 + 5 = 0
5𝑥2 + 11𝑥 + 5 = 0
5𝑥2 + 11𝑥 + 5 = 0

10. a. Express 2𝑥2 = 6 − 7𝑥 in the form of 𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0
b. Write (2x+1)(x-3) = 0 as a quadratic equation in
standard form
a. 2𝑥2
+ 7𝑥 − 6 = 0
b. 2𝑥2 − 5𝑥 − 3 = 0

11. Practice and Apply
I.Determine if the equations are quadratic or non – quadratic equations. Write Q if
the equation is a quadratic equation, write NQ if the equation is non – quadratic,
justify.
1. z2 − 2z − 1 = 0
2. 4c2 − 5c = 0
3. x3
− 8 = 0
4. h − 24h − h2 = 0
5. 4d – 3 = 10
6. 9v2
+ v = 9v2
− 6
7. 12x2 − 10x = 5x − x5
8. 5w2
− 25 = 0
9. (x+1)(x-3)=0
10. x = 2x2 + 1
11. 3-a2
= 0
12. w3 + 2 = w2
13. 3(3x+1)=0
14. x(x2
− 1) = x
15. -b2 = b − 2
NQ
NQ
NQ
NQ
NQ
Q
Q
Q
NQ
Q
Q
Q
Q
Q
NQ

12. II.Write the following quadratic equations in the form 𝑎𝑥2
+ 𝑏𝑥 + 𝑐 = 0. Identify
the values of a, b, c.
1. 6p2
= 42
2. b2 = 9 − 13b
3. t2
− 5 = 4t
4. −3x2
= 2b − 9
5. 4 – 7g − g2 = 0
6. x2
= 3x − 5
7. L=m2 − 6m
8. 2c2
= 5 − c
9. 12-f=f2
10. k2
− l = k
11. x2 = −5
12. 6x2 − 4x = 1
13.
1
2
b2 = b + 1
14. m=3-m2
15. 10=y2 − 3y

13. III.Express each of the following equations as a quadratic equation. Write your answers
in the form 𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0.
1. (2x+1)(x – 1) = 0
2. (x − 4)(2x − 5) = 0
3. x 3 − 5x = 2
4. (x+5)(3x − 1) = x(x + 1)
5. x x + 1 = x + 2
6. 3x(x+1)=(x − 2)2
7. (2x − 5)(x + 1) = 3x − 2
8. (3x − 1)2= x + 1
9. 5x = 2 + (x − 1)(x + 2)
10. 6x(3x+2)=2x(10x-1)

14. The quadratic formula is
used for finding the value
of x
a, b , c = constants, where a ≠ 0

15. 𝑥2 + 3𝑥 − 4 = 0
a = 1 b = 3 c = -4
x = 1
x +4 = 0
x = -4
By factoring: (x – 1)(x + 4) = 0
x – 1 = 0

16. 𝑥2 + 3𝑥 − 4 = 0
a = 1 b = 3 c = -4
By factoring: (x – 1)(x + 4) = 0
𝑥 =
−𝑏 ± 𝑏2 − 4𝑎𝑐
2𝑎
𝑥 =
−3 ± (3)2−4(1)(−4)
2(1)
𝑥 =
−3 ± 9 + 16
2
𝑥 =
−3 ± 25
2
𝑥 =
−3 ± 5
2
𝑥 =
−3 + 5
2
=
2
2
= 1
𝑥 =
−3 − 5
2
=
−8
2
= −4

17. 4𝑛2 + 16 = 0

18. 2𝑥2 + 4𝑥 + 8 = 24

19. 2𝑥2 − 4𝑥 − 3 = 0

20. 2𝑥2 + 5𝑥 + 3 = 0

21. 𝑥2 − 3𝑥 = 0

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