Algebra

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

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)

C. Identity elements
a² – b² = (a + b)(a – b)

Example:
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)

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

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

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

x -1 0 1 2
y -3 -8 -7 0

algebra graph - first method

from graph x = -1⅓ or 2

Second Method
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

algebra graph - second method

from graph x = -1⅓ or 2