Quadratic Functions Grade 9 ph

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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