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How to Find the Roots of a Quadratic Equation by Completing the Square

Quadratic equations are mathematical functions that take the form ax^2 + bx + c = 0, where a, b and c represent constant numbers and x is the function's independent variable. They describe the shape of parabolas, the speed of falling objects and the motion of pendulums. To solve a quadratic equation, find the values for x that result in zero. With practice, you can quickly factor some equations, such as x^2+ 2x -- 8, but not others, like x^2 + 2x - 9. For tougher cases like these, you solve using a method called "completing the square."

Instructions

- 1
  Write the equation in the standard form of ax^2 + bx + c = 0. For the example, write:
  
  x^2 + 2x - 9 = 0.
- 2
  Isolate the x^2 and x terms by subtracting the last term from both sides:
  
  x^2 + 2x -9 -(-9) = -(-9) or
  
  x^2 + 2x = 9
  
  This equation remains equivalent; you have simply rearranged it.
- 3
  Add a term to both sides equal to (b / 2)^2. In this example, b = 2, so (b / 2)^2 = 1. So you add 1 to both sides:
  
  x^2 + 2x + 1 = 9 + 1
  
  The square is now complete. x^2 + 2x + 1 on the left side is a perfect square, namely,
  
  (x + 1)^2.
- 4
  Rewrite the equation in terms of the perfect square:
  
  (x + 1)^2 = 9 + 1
  
  You can simplify this to:
  
  (x + 1)^2 = 10
- 5
  Solve the resulting equation algebraically. Take the square root of both sides:
  
  x + 1 = +/- sqrt(10)
  
  Where "sqrt(10)" means "the square root of 10." Remember, when you take the square root, the result is positive or negative. Subtracting 1 from both sides leaves x on the left side:
  
  x = -1 +/- sqrt(10). The original equation, x^2 +2x -- 9 = 0 has two roots which result in zero, namely -1 + sqrt(10) and -1 -- sqrt(10).

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