2.1-2.2: Introduction to Algebra + Manipulation:

Key Concepts

• Letters = represent generalized numbers (variables).

• Substitute numbers into expressions/formulas to evaluate.

Algebraic Manipulation

Simplify (collect like terms):

• If 2 or more numbers have the same EXACT  variable, you can simplify them

• Example: x - b + 2x = 3x - b

• Example (more details):

Expanding Products:

• Distributive property - multiply the outside term by every term inside the parentheses. After multiplying, you combine any like terms to simplify the expression

• When multiplying two sets of parentheses, such as (a + b)(c + d), every term in the first set must be multiplied by every term in the second set

Factorizing:

• This is done by taking out the common factors (numbers/variables the equation is divisible by)

• Tip: if you click on mode 5 3 (or any number in mode 5 depending on your equation style) on your calculator and embed the equation, it gives you the solutions in which you can get the factors

Special Factorization Forms:

Factorize: ax + bx + kay + kby

• This looks messy, but it’s just grouping and spotting common factors.

• Steps:

  1. Group similar terms: → (ax + bx) + (kay + kby)

  2. Factor each group: → x(a + b) + y(ka + kb)

  3. Spot the common factor: → x(a + b) + yk(a + b)

  4. Final factorization: → (a + b)(x + ky)

Factorize: a²x² − b²y²

• This is a classic difference of squares.

• Steps:

  1. Recognize the pattern: → A² − B² = (A − B)(A + B)

  2. Apply it: → a²x² − b²y² = (ax − by)(ax + by)

Factorize: a² + 2ab + b²

• This is a perfect square trinomial.

• Steps:

  1. Recognize the pattern: → A² + 2AB + B² = (A + B)²

  2. Apply it: → a² + 2ab + b² = (a + b)²

Factorize: ax² + bx + c

• This is a quadratic trinomial.

• Steps:

  1. Find two numbers that multiply to a×c and add to b.

  2. Split the middle term using those numbers.

  3. Group and factor.

• Example: Factor 2x² + 5x + 3 → a×c = 6 → 2x² + 2x + 3x + 3 → 2x(x + 1) + 3(x + 1) → (x + 1)(2x + 3)

Factorize: ax³ + bx² + cx

• This is a common factor situation.

• Steps:

  1. Factor out x: → x(ax² + bx + c)

  2. Then factor the quadratic inside (if possible): → x(…)(…)

• Example: x³ + 5x² + 6x → x(x² + 5x + 6) → x(x + 2)(x + 3)

Complete the square: ax² + bx + c

• This is a method to rewrite quadratics in a neat square form.

• Steps:

  1. If a ≠ 1, divide the whole expression by a.

  2. Take half of b/a, square it.

  3. Add and subtract that square inside the expression.

  4. Rewrite as a square.

• Example: Complete the square for 2x² + 8x + 5 

→ Divide by 2: x² + 4x + 2.5 

→ Half of 4 = 2 → 2² = 4 

→ x² + 4x + 4 − 4 + 2.5 → (x + 2)² − 1.5 

→ Final: 2(x + 2)² − 3

NOTES DONE BY FARIDA SABET

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