Completing the square

Completing the square

Basic Concepts: Factoring perfect square trinomials: (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab + b^2 or (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab + b^2, Quadratic function in vertex form: y = a(x−p)2+qa(x-p)^2 + q
Related Concepts: Solving quadratic equations by completing the square, Combining transformations of functions

Lessons

perfect squares:
  • (x+a)2=x2+2ax+a2{\left( {x + a} \right)^2} = {x^2} + 2ax + {a^2}
  • (x−a)2=x2−2ax+a2{\left( {x - a} \right)^2} = {x^2} - 2ax + {a^2}
  • completing the square: adding a constant to a quadratic expression to make it a perfect square
    • Introduction
      What is "COMPLETING THE SQUARE"?
    • 1.
      Recognizing a Polynomial that Can Be Written as a Perfect Square
      Convert the following expressions into perfect squares, if possible:
      a)
      x2+6x+32{x^2} + 6x + {3^2} =
      x2−6x+(−3)2{x^2} - 6x + {\left( { - 3} \right)^2} =

      b)
      x2+20x+100{x^2} + 20x + 100 =
      x2−20x+100{x^2} - 20x + 100 =
      x2−20x−100{x^2} - 20x - 100 =

    • 2.
      Completing the Square
      Add a constant to each quadratic expression to make it a perfect square.
      a)
      x2+10x+{x^2} + 10x + \;_____ =

      b)
      x2−2x+{x^2} - 2x + \;_____ =

      c)
      2x2+12x+2{x^2} + 12x + \;_____ =

      d)
      −3x2+60x+ - 3{x^2} + 60x + \;_____ =

      e)
      25x2−8x+\frac{2}{5}{x^2} - 8x + \;_____ =

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    14.
    Quadratic Functions
    14.1
    Characteristics of quadratic functions
    14.2
    Transformations of quadratic functions
    14.3
    Quadratic function in general form: y=ax2+bx+cy = ax^2 + bx+c
    14.4
    Quadratic function in vertex form: y = a(x−p)2+qa(x-p)^2 + q
    14.5
    Completing the square
    14.6
    Converting from general to vertex form by completing the square
    14.7
    Shortcut: Vertex formula
    14.8
    Graphing parabolas for given quadratic functions
    14.9
    Finding the quadratic functions for given parabolas
    14.10
    Applications of quadratic functions
    NZ Year 11 Maths