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Algebra: Key Concepts - Coggle Diagram

- - - - -b formula
      - graph
      - factorise
      - substitution
  - - - (b^2-4ac)
    - - (b^2-4ac) > 0
    - - (b^2-4ac) < 0
    - - = to a perfect square
    - - (b^2-4ac) = 0
  - - - Isolate
        Square
        Solve
        Substitute
    - - Move one to each side
        Square both sides
        Isolate any remaining surds
        Square again
        Solve
        Substitute
  - - - Trial and error in f(x) until it = 0 and divide
    - - Find factors, equate to 0 and then find roots
- - - - less than/equal to ≤
      - more than/equal to ≥
      - less than <
      - more than >
    - - ax^2 + bx + c ≥ 0
        
        solve to find real roots, draw a rough sketch, use graph to find set of values that satisfy the inequality
      - Graph
        
        If a > 0, graph is U-shaped
        
        If a < 0, graph is ⋂-shaped
    - - (( 3x - 2 ) / (x + 1 )) ≥ 0
        
        Multiply both sides by (x + 1)^2 to retain inequality as any number squared is always positive ----> multiply out both sides and solve quadratic inequality
  - - - If ∣x∣ < 1 then x lies between 1 and -1.... i.e -1 < x < 1
      - If ∣x∣ > 1 then x lies outside.... i.e x > 1 or x < -1
      - Middle Test
        
        To check whether x is inside or outside, get the middle number between the two x values of a quadratic and sub in
  - - - Axioms, definitions and other proven theorems are used to verify statements
    - - We show that if a statement is to be true, then a logical contradiction occurs and so it proves the original statement to be false
      - Prove that √2 is irrational
        
        Assume it's rational, if rational then it can be written as a/b (a,b ∈ Z) and b=/= 0, and a and b have no common factors)
        
        square both sides, a^2 is even since it's equal to 2b^2, a is even, b is even, let a = 2c
        
        CONTRADICTION: a and b have common factor 2 so √2 is irrational
  - - - 4 = base number
      - 2 = power/index
  - - - a^x = y is the same as loga y = x
      - the log of a number to its own base = 1
    - - loge = natural
      - log10 = common
    - - check that each term has the same base, if not, the change of base rule must be used to change to a common base,,, check solutions, as negative logs aren't defined