## 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 | |

Inequalityis a mathematical sentence that uses the sign of inequality |
4 < 6 |

24 + 4 > 20 + 2 | |

6 ≠ 4 | |

Equationopen sentence which states the relationship “equal to” |
x + 3 = 5 |

2 x² – 7x = 15 | |

Inequalityopen 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 + 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

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

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:

y = 3x² – 2x – 8

x |
-1 | 0 | 1 | 2 |

y |
-3 | -8 | -7 | 0 |

from graph x = -1⅓ or 2

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