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Quadratic Formula - Coggle Diagram

- - - - Formula: x = (-b ± √(b² - 4ac)) / 2a
      - Discriminant (Δ = b² - 4ac)
        
        Δ > 0 → 2 real and distinct roots
        
        Δ = 0 → 1 real and repeated root
        
        Δ < 0 → 2 complex roots
      - Works for all quadratic equations
      - Example:
        
        2x² + 3x - 2 = 0
        
        Δ = 3² - 4(2)(-2) = 9 + 16 = 25
        
        x = (-3 ± √25) / 4
        
        x = (-3 ± 5) / 4 → x = 1/2 or -2
    - - Express quadratic as (x + m)(x + n) = 0
      - Steps:
        
        Multiply a × c (from ax² + bx + c)
        
        Find two numbers that multiply to ac and add to b
        
        Rewrite and group terms
        
        Factor by grouping
      - Works well when coefficients are simple
      - Example:
        
        x² + 5x + 6 = 0
        
        Factors of 6 that sum to 5 → 2 and 3
        
        (x+2)(x+3) = 0
        
        Roots: x = -2, -3
    - - Convert expression into a perfect square trinomial
      - Steps:
        
        Divide through by a (if a ≠ 1)
        
        Take half of coefficient of x, square it
        
        Add and subtract this value inside equation
        
        Write as (x + p)² = q
        
        Solve for x by square root
      - Example:
        
        x² + 6x + 5 = 0
        
        x² + 6x = -5
        
        (b/2)² = (6/2)² = 9
        
        Add 9 to both sides: x² + 6x + 9 = 4
        
        (x+3)² = 4
        
        x+3 = ±2 → x = -1, -5
      - Important:
        
        Basis for deriving quadratic formula