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Math Help - Question on Completing the X

  1. #1
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    Question on Completing the X

    Hello there, I have a question on how to complete the x on a quadratic equation which has the coefficient of a. It cannot be solved using factorization. The equation is:

    8x^2 + 4x - 1 = 0

    Thank You
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  2. #2
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    I assume you mean to complete the square...

    8x^2 + 4x - 1 = 0

    8\left(x^2 + \frac{1}{2}x - \frac{1}{8}\right) = 0

    8\left[x^2 + \frac{1}{2}x + \left(\frac{1}{4}\right)^2 - \left(\frac{1}{4}\right)^2 - \frac{1}{8}\right] = 0

    8\left[\left(x + \frac{1}{4}\right)^2 - \frac{3}{16}\right] = 0

    8\left(x + \frac{1}{4}\right)^2 - \frac{3}{2} = 0.


    Go from here.
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  3. #3
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    Quote Originally Posted by Prove It View Post
    I assume you mean to complete the square...

    8x^2 + 4x - 1 = 0

    8\left(x^2 + \frac{1}{2}x - \frac{1}{8}\right) = 0

    8\left[x^2 + \frac{1}{2}x + \left(\frac{1}{4}\right)^2 - \left(\frac{1}{4}\right)^2 - \frac{1}{8}\right] = 0

    8\left(x + \frac{1}{4}\right)^2 - \frac{3}{16}\right] = 0

    8\left(x + \frac{1}{4}\right)^2 - \frac{3}{2} = 0.

    Go from here.
    Thank you but I'm little bit confused on how do you exactly have

    - \frac{3}{2}

    From

    \frac{3}{16}

    I've also noticed that the second from the last line did not show up, so I edit your formatting a little bit. I'm not sure if what I'm doing is right. Can you verify?
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  4. #4
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    I expanded the outermost (square) brackets.

    8\left(-\frac{3}{16}\right) = -\frac{3}{2}.
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  5. #5
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    Quote Originally Posted by Prove It View Post
    I expanded the outermost (square) brackets.

    8\left(-\frac{3}{16}\right) = -\frac{3}{2}.
    Ah I see, can you give me a bit of insight how to find x? Or at least to eliminate the 8?
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  6. #6
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    Use the standard method of "undoing" everything using inverse operations.
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  7. #7
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    Is it possible to do this:

    8(x+1/4)^2 = 3/2

    8(x+1/4) = root(3/2)

    8x + 2 = root(3/2)

    8x = root(3/2) - 2

    x = (root(3/2) - 2) / 8

    But it gives up quite weird answer...

    I'm sorry also because of bad formattig I do not know how to operate Latex properly.
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  8. #8
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    No, when you undo everything you go in the reverse of the order of operations.

    Since Exponentiation comes before Multiplication, when you go in reverse, the Multiplication is undone before the Exponentiation...
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  9. #9
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    Quote Originally Posted by Prove It View Post
    No, when you undo everything you go in the reverse of the order of operations.

    Since Exponentiation comes before Multiplication, when you go in reverse, the Multiplication is undone before the Exponentiation...
    Allright, would this be possible:



    8(x+1/4)^2 = 3/2

    (x+1/4)^2 = 8(3/2)

    (x+1/4) = root(12)

    x = root(12)(1/4)
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  10. #10
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    How do you undo multiplication?
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  11. #11
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    Quote Originally Posted by Prove It View Post
    How do you undo multiplication?
    By dividing it I suppose, I'm really sorry if action looks like an imbecible. I just want to understand it. Just having problem with that number 8.
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  12. #12
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    Yes, you need to divide both sides by 8.

    What is \frac{3}{2} \div 8?
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  13. #13
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    Quote Originally Posted by Prove It View Post
    Yes, you need to divide both sides by 8.

    What is \frac{3}{2} \div 8?
    AH I see would it be like this then?

    (x+\frac{1}{4})^2 = \frac{3}{16}
    (x+\frac{1}{4}) = \sqrt{\frac{3}{16}}
    (x+\frac{1}{4}) = \frac{\sqrt{3}}{4}
    x = \frac{1}{4}(1-\sqrt{3})
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  14. #14
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    Quote Originally Posted by Lites View Post
    AH I see would it be like this then?

    (x+\frac{1}{4})^2 = \frac{3}{16}
    (x+\frac{1}{4}) = \sqrt{\frac{3}{16}}
    (x+\frac{1}{4}) = \frac{\sqrt{3}}{4}
    x = \frac{1}{4}(1-\sqrt{3})
    Very close. There are two square roots to every nonnegative number, one positive and one negative.

    So that means it should read...

    \left(x + \frac{1}{4}\right)^2 = \frac{3}{16}

    x + \frac{1}{4} = \pm \sqrt{\frac{3}{16}}

    x + \frac{1}{4} = \pm \frac{\sqrt{3}}{4}

    x = -\frac{1}{4} \pm \frac{\sqrt{3}}{4}.


    So your two solutions are x = \frac{-1 - \sqrt{3}}{4} and x = \frac{-1 + \sqrt{3}}{4}.
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  15. #15
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    Quote Originally Posted by Prove It View Post
    Very close. There are two square roots to every nonnegative number, one positive and one negative.

    So that means it should read...

    \left(x + \frac{1}{4}\right)^2 = \frac{3}{16}

    x + \frac{1}{4} = \pm \sqrt{\frac{3}{16}}

    x + \frac{1}{4} = \pm \frac{\sqrt{3}}{4}

    x = -\frac{1}{4} \pm \frac{\sqrt{3}}{4}.


    So your two solutions are x = \frac{-1 - \sqrt{3}}{4} and x = \frac{-1 + \sqrt{3}}{4}.
    Finally, I got it!

    Thank you for your kind help Prove it!
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