Completing the Square

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Completing the Square

  1. 1. Solving Quadratic Equations By Completing the Square
  2. 2. What is a Perfect Square Trinomial?
  3. 3. How to create a Perfect Square Trinomial <ul><li>In the following perfect square </li></ul><ul><li>trinomial, the constant term is missing. </li></ul><ul><li>x 2 + 14x + ____ </li></ul><ul><li>Find the constant term by squaring half </li></ul><ul><li>the coefficient of the linear term. </li></ul><ul><li>(14/2) 2 </li></ul><ul><li>X 2 + 14x + 49 </li></ul>
  4. 4. Perfect Square Trinomials 100 4 25/4
  5. 5. How to “Complete the Square” Solve the following equation by completing the square: x 2 + 8x – 20 = 0 Step 1: Move quadratic term, and linear term to left side of the equation x 2 + 8x = 20
  6. 6. Step 2: Find the term that completes the square on the left side of the equation. Add that term to both sides. x 2 + 8x + ? = 20 + ? x 2 + 8x + 16 = 20 + 16 8/2 = 4 4 2 = 16
  7. 7. Step 3: Factor the perfect square trinomial on the left side of the equation. Simplify the right side of the equation.
  8. 8. Step 4: Take the square root of each side and solve. x = 10 or x = -2
  9. 9. Example #2 2 2 2
  10. 10. Step 2: Find the term that completes the square on the left side of the equation. Add that term to both sides. x 2 – 7/2 + 49/16 = -6 + 49/16 x 2 – 7/2 + 49/16 = -96/16 + 49/16 x 2 – 7/2 + 49/16 = -47/16 ½ * 7/2 = 7/4 (7/4) 2 = 49/16
  11. 11. <ul><li>x 2 – 7/2 + 49/16 = -47/16 </li></ul><ul><li>(x -7/2)(x -7/2) = -47/16 </li></ul><ul><li>(x – 7/2) 2 = -47/16 </li></ul><ul><li>(x – 7/2) 2 = -47/16 </li></ul><ul><li>x -7/2 = i 47 </li></ul>Step 3: Factor the perfect square trinomial on the left side of the equation Step 4: Take the square root of each side 4 X = 7/2 + i 47 or x = 7/2 - i 47 4 4

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