Completing the square

  1. Recognizing a Polynomial that Can Be Written as a Perfect Square
    Convert the following expressions into perfect squares, if possible:
    1. x2+6x+32{x^2} + 6x + {3^2} =
      x26x+(3)2{x^2} - 6x + {\left( { - 3} \right)^2} =
    2. x2+20x+100{x^2} + 20x + 100 =
      x220x+100{x^2} - 20x + 100 =
      x220x100{x^2} - 20x - 100 =
  2. Completing the Square
    Add a constant to each quadratic expression to make it a perfect square.
    1. x2+10x+  {x^2} + 10x + \;_____ =
    2. x22x+  {x^2} - 2x + \;_____ =
    3. 2x2+12x+  2{x^2} + 12x + \;_____ =
    4. 3x2+60x+   - 3{x^2} + 60x + \;_____ =
    5. 25x28x+  \frac{2}{5}{x^2} - 8x + \;_____ =
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Topic Notes
perfect squares:
  • (x+a)2=x2+2ax+a2{\left( {x + a} \right)^2} = {x^2} + 2ax + {a^2}
  • (xa)2=x22ax+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