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3.5 – Completing the Square


  • Complete the square algebraically to find the solutions to a quadratic equation.
  • Complete the square with tiles.
  • Determine the number that must be added to an expression to complete the square.


Key Terms

  • Complete the Square – A way to solve quadratic equations. It involves adding a number to both sides of an equation to make one side a perfect square trinomial.
    • The equation can then be solved by taking the square root of both sides and simplifying.
  • Perfect Square Trinomial – The result of multiplying a binomial by itself (squaring it).
    • Ex. x^2-10x+100=(x-10)(x-10)=(x-10)^2



Solving Quadratics with Perfect Square Trinomials
  • How to Solve a Quadratic Equation with a Perfect Square Trinomial
    • Step 1: Factor the perfect square trinomial.
    • Step 2: Take the square root of each side of the equation.
    • Step 3: Simplify the resulting equations.
    • Step 4: Solve for x in each equation.

Alg2A 3.05 - CompleteSqare02

Completing the Square
  • Formula:  Add (\frac{b}{2})^2 to both sides

Alg2A 3.05 - CompleteSquareRule

  • Remember: “b” is the middle term in standard form (the x-term)

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  • Remember: what you do to one side, you MUST do to the other (to balance the equation)Alg2A 3.05 - CompleteSqare03
  • Algebra Tiles – A visual model of a “Perfect Square” trinomial

Alg2A 3.05 - CompleteSqare04visual

  • Example: x^2=100 has 2 solutions (one positive and one negative)
    • (\sqrt{x})^2=x and \sqrt{100}=\pm10
    • So, x = 10 or x = -10
Alg2A 3.05 - CompleteSquare01

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