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 =(x+3)2 =(x-5)2 =(x+6)2 x2 + 6x + 9 x2 - 10x + 25 x2 + 12x + 36 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 x2 - 10x + 25 Half of these values squared … 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

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.

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. 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

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.

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.

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! Use calculator to do this! Use calculator to do this! Use calculator to do this! Use calculator to do this!