Completing the Square for Conic Sections The Aim of Completing the Square … is to write a quadratic function as a perfect square. Here are some examples of perfect squares! x2 + 6x + 9 x2 - 10x + 25 x2 + 12x + 36 Try to factor these (they’re easy). Perfect Square Trinomials x2 + 6x + 9 x2 - 10x + 25 x2 + 12x + 36 =(x+3)2 =(x-5)2 =(x+6)2 Can you see a numerical connection between … 6 and 9 using 3 -10 and 25 using -5 12 and 36 using 6 The Perfect Square Connection For a perfect square, the following relationships will always be true … x2 + 6x + 9 Half of these values squared x2 - 10x + 25 … are these values The Perfect Square Connection In the following perfect square trinomial, the constant term is missing. Can you predict what it might be? X2 + 14x + ____ Find the constant term by squaring half the coefficient of the linear term. (14/2)2 X2 + 14x + 49 Perfect Square Trinomials Create perfect square trinomials. x2 + 20x + ___ x2 - 4x + ___ x2 + 5x + ___ 100 4 25/4 Solving Quadratic Equations by Completing the Square Solve the following equation by completing the square: Step 1: Move the constant term (i.e. the number) to right side of the equation x 8 x 20 0 2 x 8 x 20 2 Solving Quadratic Equations by Completing the Square Step 2: Find the term that completes the square on the left side of the equation. Add that term to both sides. x 8x 2 =20 + 1 (8) 4 then square it, 42 16 2 x 8 x 16 20 16 2 Solving Quadratic Equations by Completing the Square Step 3: Factor the perfect square trinomial on the left side of the equation. Simplify the right side of the equation. x 8 x 16 20 16 2 ( x 4)( x 4) 36 ( x 4) 36 2 For chapter 10 material, we can stop here. But solving is a simple process from here … Solving Quadratic Equations by Completing the Square Step 5: Set up the two possibilities and solve x 4 6 x 4 6 and x 4 6 x 10 and x=2 Completing the Square-Example #2 Solve the following equation by completing the square: Step 1: Move the constant to the right side of the equation. 2 x 7 x 12 0 2 2 x 7 x 12 2 Solving Quadratic Equations by Completing the Square Step 2: Find the term that completes the square on the left side of the equation. Add that term to both sides. The quadratic coefficient must be equal to 1 before you complete the square, so you must divide all terms by the quadratic coefficient first. 2 x2 7 x =-12 + 2 x2 7 x 12 2 2 2 2 7 x x 6 2 1 7 7 ( ) then square it, 2 2 4 7 49 49 x x 6 2 16 16 2 2 7 49 4 16 Solving Quadratic Equations by Completing the Square Step 3: Factor the perfect square trinomial on the left side of the equation. Simplify the right side of the equation. Use calculator to do this! 7 49 49 x x 6 2 16 16 2 2 7 96 49 x 4 16 16 2 7 47 x 4 16

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# The Aim of Completing the Square