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Completing the Square

Completing the Square. Grade 10 Lesson 5-5. Completing the Square. This is an x. Show me x 2. x 2. x. Show me x 2 + 6x. Completing the Square. x. x. x. x. x. x. x 2. x 2. Let's Make a Square. Completing the Square. How many units are required to complete the square?.

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Completing the Square

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  1. Completing the Square Grade 10 Lesson 5-5

  2. Completing the Square • This is an x. Show mex2. x2 x Show me x2 + 6x

  3. Completing the Square x x x x x x x2 x2 Let's Make a Square

  4. Completing the Square How many units are required to complete the square? 9! The picture is now x2 + 6x + 9, which factors (x+3)(x+3) = (x+3)2

  5. Let’s Try Another One! • Show me x2 + 2x Let's Make a Square x2 x x

  6. Completing the Square How many units are required to complete the square? 1! x2 x2 The picture is now x2 + 2x + 1 which factors (x+1)(x+1) = (x+1)2

  7. Last One with Manipulative Show me x2 + 8x Let's Make a Square x x x x x x x x x2

  8. Completing the Square Again, how many units are required to complete the square? 16 So,the picture is now x2 + 8x + 16 which factors (x+4)(x+4) = (x+4)2

  9. Hard One! Complete the square for x2 + 18x + ___ How many units are needed? There are not enough pieces to do this problem. Can we do it using paper and pencil?

  10. What is completing the square used for? • Completing the square is used for all those not factorable problems!! • It is used to solve these equations for the variable.

  11. Rule for Completing the Square This is now a PTS! So, it factors into this!

  12. Example: Find the value of c that makes this a PTS, then write the expression as the square of a binomial. x2-3x+c b=-3

  13. Example: Solve by completing the square. x2+6x-8=0 x2+6x-8=0 x2+6x=8 x2+6x+___=8+___ x2+6x+9=8+9 (x+3)2=17 Don’t forget: Whatever you add to one side of an equation, you MUST add to the other side!

  14. 5x2-10x+30=0 x2-2x+6=0 x2-2x=-6 x2-2x+__=-6+__ x2-2x+1=-6+1 (x-1)2=-5 3x2-12x+18=0 x2-4x+6=0 x2-4x=-6 x2-4x+__=-6+__ x2-4x+4=-6+4 (x-2)2=-2 More Examples!

  15. y=x2+6x+16 y-16=x2+6x y-16+__=x2+6x+__ y-16+9=x2+6x+9 y-7=(x+3)2 y=(x+3)2+7 If the equation, in vertex form, is y=(x+3)2+7, then the vertex must be (-3,7). Last Example! Write the quadratic function y=x2+6x+16 in vertex form. What is the vertex of the function’s graph?

  16. Solving Quadratic Equations by Completing the Square

  17. Perfect Square Trinomials • Examples • x2 + 6x + 9 • x2 - 10x + 25 • x2 + 12x + 36

  18. Creating a Perfect Square Trinomial • In the following perfect square trinomial, the constant term is missing. X2 + 14x + ____ • Find the constant term by squaring half the coefficient of the linear term. • (14/2)2 X2 + 14x + 49

  19. Perfect Square Trinomials • Create perfect square trinomials. • x2 + 20x + ___ • x2 - 4x + ___ • x2 + 5x + ___ 100 4 25/4

  20. Solving Quadratic Equations by Completing the Square Solve the following equation by completing the square: Step 1: Move quadratic term, and linear term to left side of the equation

  21. 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.

  22. 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.

  23. Solving Quadratic Equations by Completing the Square Step 4: Take the square root of each side

  24. Solving Quadratic Equations by Completing the Square Step 5: Set up the two possibilities and solve

  25. Completing the Square-Example #2 Solve the following equation by completing the square: Step 1: Move quadratic term, and linear term to left side of the equation, the constant to the right side of the equation.

  26. 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.

  27. 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.

  28. Solving Quadratic Equations by Completing the Square Step 4: Take the square root of each side

  29. Solving Quadratic Equations by Completing the Square Try the following examples. Do your work on your paper and then check your answers.

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