Page 8 - TOP ONE ADDMATHS TG4

Additional Mathematics Form 4 Chapter 2 Quadratic Functions

2.2 Types of Roots of Quadratic Equations Textbook
Jenis-jenis Punca Persamaan Kuadratik
pg. 45 – 48

SMART Notes

1. The type of roots of a quadratic equation 2. Two different real roots
2
ax + bx + c = 0 can be determined by finding the b – 4ac . 0 Dua punca nyata yang berbeza
2
value of discriminant, b – 4ac.
2
Jenis-jenis punca bagi suatu persamaan kuadratik b – 4ac = 0 Two equal real roots
2
ax + bx + c = 0 boleh ditentukan dengan mencari nilai Dua punca nyata yang sama
2
pembezalayan, b – 4ac. No real roots
2
b – 4ac , 0
2
Tiada punca nyata
7. Find the value of discriminant for each of the following quadratic equations. Then, determine the type of
roots of the quadratic equation. PL 3
Cari nilai pembezalayan bagi setiap persamaan kuadratik berikut. Kemudian, tentukan jenis punca bagi persamaan kuadratik tersebut.
Example (a) x – 8x = –16
2
(x + 2) = 12x – 11
2
x – 8x + 16 = 0
2
x + 4x + 4 = 12x – 11
2
x – 8x + 15 = 0 Discriminant, b – 4ac
2
2
= (–8) – 4(1)(16)
2
Discriminant, b – 4ac = 0
2
2
= (–8) – 4(1)(15)
= 4 . 0
∴ two equal real roots
∴ The equation has two different real roots.
1
(b) x(6x – 5) = 1 (c) x + 2 = – x
2
2
6x – 5x – 1 = 0 2(x + 2) = –x
2
2
2x + 4 = –x
2
Discriminant, b – 4ac 2x + x + 4 = 0
2
2
2
= (–5) – 4(6)(–1)
2
= 49 . 0 Discriminant, b – 4ac
= (1) – 4(2)(4)
2
∴ two different real roots = –31 , 0
∴ no real roots
8. Find the range of values of p if each of the following quadratic equations has two different real roots. PL 3
Cari julat nilai p jika setiap persamaan kuadratik berikut mempunyai dua punca nyata yang berbeza.
Example (a) 4x – 5x + 3p – 1 = 0 (b) x + px + 4 = 0
2
2
3x – 12x + p = 0 For two different real roots:
2
For two different real roots: b – 4ac . 0
2
b – 4ac . 0
2
For two different real roots: 2 (p) – 4(1)(4) . 0
2
b – 4ac . 0 (–5) – 4(4)(3p – 1) . 0 p – 16 . 0
2
2
(–12) – 4(3)(p) . 0 25 – 16(3p – 1) . 0 (p – 4)(p + 4) . 0
2
144 – 12p . 0 25 – 48p + 16 . 0
–12p . –144 41 – 48p . 0 When (p – 4)(p + 4) = 0,
12p , 144 –48p . –41 p = 4 or p = –4
p , 12 48p , 41
p , 41
48 p
–4 4
∴ p , –4 or p . 4
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