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Math Help - completing square

  1. #1
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    completing square

    dear,

    anybody knws how is the easiest way to completing square? or a more simple way?
    ex : x^2 - 4x - 5 = 0

    thank you
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  2. #2
    Super Member Deadstar's Avatar
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    Note that if you expand (x + a)^2 you get x^2 + 2ax + a^2. Now look at the 'a' value in the 2ax term. That is your 'a' value in the (x + a)^2 expression.

    So if you have a equation such as x^2 + 6x + 6... This can be re-written as x^2 + 2 \cdot 3x + 6 and hence 3 is your 'a' value. So in this example completing the square would be...

    x^2 + 6x + 6 = (x + 3)^2 - 3 since expanding (x + 3)^2 gives x^2 + 6x + 9 so you need to take away a 3 from it to match your original equation.

    Basically to generalize... If you have an equation of the form x^2 + nx + m where n and m are some numbers, you should divide n by 2 to get the 'a' value in the equation (x + a)^2.

    So lets do your example...
    x^2 - 4x - 5 = 0
    Your n value is -4. Divide this by 2 to get -2. This is your 'a' value. Put this into the equation (x + a)^2 to get... (x - 2)^2.

    Now expand this... (x - 2)^2 = x^2 - 4x + 4. Since this is different from you original equation you need to subtract 9 to turn the +4 into a -9...

    So your answer is! (x - 2)^2 - 9.
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  3. #3
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    Completing the Square

    Hello janeyandtaz
    Quote Originally Posted by janeyandtaz View Post
    dear,

    anybody knws how is the easiest way to completing square? or a more simple way?
    ex : x^2 - 4x - 5 = 0

    thank you
    Completing the square is a bit tricky to explain, but you may find you can follow the technique I've described here, where I've broken it down to 6 steps in my answer to Question 1. (You'll see that I've given an example where the coefficient of x is not equal to 1.)

    Grandad
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