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Thread: completing the square

  1. #1
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    completing the square

    Hi;
    is it possible to complete the square on a perfect square trinomial?
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  2. #2
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    Re: completing the square

    Quote Originally Posted by anthonye View Post
    Hi;
    is it possible to complete the square on a perfect square trinomial?
    What? Like $(a+b+c)^2$? Isn't that already "complete"?
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  3. #3
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    Re: completing the square

    just wandered if and how?
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  4. #4
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    Re: completing the square

    Quote Originally Posted by anthonye View Post
    just wandered if and how?
    Yes, and by repeating exactly what you are given. If you have $(a+b+c)^2$, then it is already complete. There is no need to "complete" it further. If that is not what you mean, then I am not understanding what you do mean.
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  5. #5
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    Re: completing the square

    ok so if its already complete there is no need to complete it further... But can you?
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  6. #6
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    Re: completing the square

    Quote Originally Posted by anthonye View Post
    ok so if its already complete there is no need to complete it further... But can you?
    I don't understand what you are asking. Do you have an example?

    This is the best example I could come up with:

    If $a,b,f\ge 0$:

    $$ax^2+by^2+cxy+dx+ey+f = (gx+hy+i)^2+jxy+kx+ly$$

    Multiplying out gives:

    $$ax^2+by^2+cxy+dx+ey+f = g^2x^2+h^2y^2+2ghxy+2gix+2hiy+i^2+jxy+kx+ly$$

    Comparing like terms, we have:

    $$g^2=a \Longrightarrow g=\sqrt{a}$$

    $$h^2=b \Longrightarrow h=\sqrt{b}$$

    $$2gh+j = c \Longrightarrow j=c-2\sqrt{ab}$$

    $$i^2 = f \Longrightarrow i=\sqrt{f}$$

    $$2gi+k = d \Longrightarrow k=d-2\sqrt{af}$$

    $$2hi+l = e \Longrightarrow l=e-2\sqrt{bf}$$

    Is that what you mean? Because this is one of many possible solutions.

    In general, you have:

    $$ax^2+by^2+cxy+dx+ey+f = (gx+hy+i)^2+jxy+kx+ly+m$$

    But, this gives six coefficients on the LHS with seven variables on the RHS (in other words, you have to choose a value for one of the RHS coefficients). In my example above, I chose $m=0$. I could just as easily have chosen the value for $g, h, i, j, k,$ or $l$. That is what I mean when I say there are many ways this could be done.
    Last edited by SlipEternal; Nov 1st 2018 at 02:06 PM.
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  7. #7
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    Re: completing the square

    Ok this is a perfect t square trinomial 4x^2 + 12x + 9 can I complete the square on it?
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  8. #8
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    Re: completing the square

    Ooh! That is totally different from what I was describing.

    $$4x^2+12x+9=(ax+b)^2=a^2x+2abx+b^2$$

    By equating coefficients on the LHS and RHS, we have $a^2=4, 2ab=12, b^2=9$. This implies $a=\pm 2, b=\pm 3$. So there are two solutions: $4x^2+12x+9=(2x+3)^2=(-2x-3)^2$
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  9. #9
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    Re: completing the square

    Quote Originally Posted by anthonye View Post
    ok so if its already complete there is no need to complete it further... But can you?
    You are asking "Can you complete something that is already completed"! What does that even mean?
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