1. Mathematical Sentence
Mathematical sentences include:
| Type of Sentences | Example |
| Correct sentence | 2 + 3 = 5 |
| 3 x 0² – 2 x – 8 = 0 | |
| x = 2 or x = -1⅓ | |
| Wrong sentence | 5 – 3 = 3 |
| 12 + 4x – x² = (2 + x) (x – 5) | |
| Open Sentence | x + 2 = 6 |
| y = x² – 4 | |
| similarities are open sentences that contain signs of similarity | 6 + 2 = 8 |
| 12 – 6 = 6 | |
| Inequality is a mathematical sentence that uses the sign of inequality |
4 < 6 |
| 24 + 4 > 20 + 2 | |
| 6 ≠ 4 | |
| Equation open sentence which states the relationship “equal to” |
x + 3 = 5 |
| 2 x² – 7x = 15 | |
| Inequality open sentences that use a sign of inequality and contain variables |
x + 3 < 6 |
| (x – 5) (x – 8) ≤ 0 | |
| 3 + 5x – 2x² > 0 |
2. Factorial
A. by finding equation factor
Example:
6a³b – 2a² b + 8 ab = 2 ab (3a² – a + 4)
2 ab is equation factor
6a³b – 2a² b + 8 ab = 2 ab (3a² – a + 4)
2 ab is equation factor
B. Collecting and Grouping
Example:
2px – 3qy – qx + 6py =
(2px – qx) + (6py – 3qy) =
x (2 p – q) + 3 y (2p – q) =
(2 p – q) (x + 3y)
2px – 3qy – qx + 6py =
(2px – qx) + (6py – 3qy) =
x (2 p – q) + 3 y (2p – q) =
(2 p – q) (x + 3y)
C. Identity elements
a² – b² = (a + b)(a – b)
Example:
16 c² – 9 d² = 4²c² – 3²d²
= (4c + 3d)(4c – 3d)
16 c² – 9 d² = 4²c² – 3²d²
= (4c + 3d)(4c – 3d)
D. Factoring Quadratic
Example: x² + 5x + 6
Result of times 6x²
Total 5x
Obtained: 2x + 3x
x² + 5x + 6 can be converted into x² + 2x + 3x + 6
= (x² + 2x) + (3x + 6)
= x (x + 2) + 3 (x + 2)
= (x + 3)(x + 2)
= (x² + 2x) + (3x + 6)
= x (x + 2) + 3 (x + 2)
= (x + 3)(x + 2)
3. Quadratic Equations
ax² + bx + c = 0 with a ≠ 0
a, b, c ϵ R
is a quadratic equation.
The solution can be done with:
a. Factorial
Example:
2a² – 5a + 3 = 0
2a² – 2a – 3a + 3 = 0
(2a² – 2a) – (3a – 3) = 0
2a (a – 1) – 3(a – 1) = 0
(2a – 3)(a – 1) = 0
2a = 3 or a = 1
a = 1½ or a = 1
2a² – 5a + 3 = 0
2a² – 2a – 3a + 3 = 0
(2a² – 2a) – (3a – 3) = 0
2a (a – 1) – 3(a – 1) = 0
(2a – 3)(a – 1) = 0
2a = 3 or a = 1
a = 1½ or a = 1
b. Complete Quadratic
ax² + bx + c = 0
– move c to the right side
ax² + bx = -c
– add the two sections with (½ coefficient x)²
ax² + bx + (½b)² = -c + (½b)²
– then factorized
Example:
x² + 2x – 8 = 0
x² + 2x = 8
x² + 2x + (2/2)² = 8 + (2/2)²
x² + 2x + 1 = 8 + 1
(x + 1)² = 9
x + 1 = √9
x + 1 = ± 3
x = 2 or x = -4
x² + 2x – 8 = 0
x² + 2x = 8
x² + 2x + (2/2)² = 8 + (2/2)²
x² + 2x + 1 = 8 + 1
(x + 1)² = 9
x + 1 = √9
x + 1 = ± 3
x = 2 or x = -4
c. Using Formula
d. Using Graph
for Example:
3x² – 2x – 8 = 0
this problem can be solved in 2 method:
First method
y = 3x² – 2x – 8
y = 3x² – 2x – 8
| x | -1 | 0 | 1 | 2 |
| y | -3 | -8 | -7 | 0 |
from graph x = -1⅓ or 2
Second Method
3x² – 2x – 8 = 0
3x² = 2x + 8
y1 = 3x²
3x² – 2x – 8 = 0
3x² = 2x + 8
y1 = 3x²
| x | -2 | -1 | 0 | 1 | 2 |
| y | 12 | 3 | 0 | 3 | 12 |
y2 = 2x + 8
| x | -1 | 0 | 1 |
| y | 6 | 8 | 10 |
from graph x = -1⅓ or 2
