Solving Quadratics by Completing the Square

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Solving Quadratics by Completing the Square

  1. 1. Solving Quadratic Equations by Completing the Square
  2. 2. Perfect Square Trinomials Examples x2 + 6x + 9 x2 - 10x + 25 x2 + 12x + 36
  3. 3. 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
  4. 4. Perfect Square Trinomials Create perfect square trinomials. x2 + 20x + ___ x2 - 4x + ___ x2 + 5x + ___ 100 4 25/4
  5. 5. 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 x + 8 x − 20 = 0 2 x + 8 x = 20 2
  6. 6. 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 W=20 +W 1 • (8) = 4 then square it, 42 = 16 2 x + 8 x + 16 = 20 + 16 2
  7. 7. 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
  8. 8. Solving Quadratic Equations by Completing the Square Step 4: Take the square root of each side ( x + 4) = 36 2 ( x + 4) = ±6
  9. 9. 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
  10. 10. 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. 2 x − 7 x + 12 = 0 2 2 x − 7 x = −12 2
  11. 11. Solving Quadratic Equations by Completing the Square 2x − 7x + W =-12 +W Step 2: Find the term 2x 7x 12 that completes the square − + W= − + W on the left side of the 2 2 2 2 2 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. 7 x − x +W −6 +W = 2 2 1 7 7 • ( ) = then square it, 2 2 4 7 49 49 x − x+ = −6 + 2 16 16 2 2 49 7  ÷ =  4  16
  12. 12. 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. 7 49 49 x − x+ = −6 + 2 16 16 2 2 7 96 49  x− ÷ =− + 4 16 16  2 7 47  x− ÷ =− 4 16 
  13. 13. Solving Quadratic Equations by Completing the Square Step 4: Take the square root of each side 7 2 −47 (x − ) = 4 16 7 −47 (x − ) = ± 4 4 7 i 47 x= ± 4 4 7 ± i 47 x= 4
  14. 14. Solving Quadratic Equations by Completing the Square Try the following examples. Do your work on your paper and then check your answers. 1. x + 2 x − 63 = 0 2 2. x + 8 x −84 = 0 2 3. x − 5 x − 24 = 0 2 4. x + 7 x +13 = 0 2 5. 3 x 2 +5 x + 6 = 0 1. ( −9, 7 ) 2.(6, −14) 3. ( −3,8 )  −7 ± i 3  4.  ÷  ÷ 2    −5 ± i 47  5.  ÷  ÷ 6  

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