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Math Help - Is this the discriminate?

  1. #1
    Senior Member Mukilab's Avatar
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    Is this the discriminate?

    My equation is x^2+2kx-5=0

    I have to find the discriminate for this assuming it has 2 real roots

    so I do b^2-4ac is greater than 0

    (2k)^2-4(-5) is greater than 0
    k^2 is greater than -5
    k= plus or minus square route of -5


    I seem to have done the discriminate well but how can a complex number be the solution for the discriminate since I am supposed to have 2 real roots?

    I also encountered the same problem trying to do x^2+2kx-7=0 (but with this time the square route of -7). Can anyone help out please? Thanks.
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    Re: Is this the discriminate?

    Quote Originally Posted by Mukilab View Post
    My equation is x^2+2kx-5=0
    I have to find the discriminate for this assuming it has 2 real roots
    so I do b^2-4ac is greater than 0
    (2k)^2-4(-5) is greater than 0
    k^2 is greater than -5
    k= plus or minus square route of -5
    I seem to have done the discriminate well but how can a complex number be the solution for the discriminate since I am supposed to have 2 real roots?
    I also encountered the same problem trying to do x^2+2kx-7=0 (but with this time the square route of -7). Can anyone help out please? Thanks.
    The discriminate is 4k^2+20 which is positive if k\ne 0~.
    So it has two real root for any k~.
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  3. #3
    Senior Member Mukilab's Avatar
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    Re: Is this the discriminate?

    Can you please explain how to get those real roots? The method that I learned from my book clearly isn't working.

    My method:
    I got the discriminate with no problem
    but then I equalled it to 0 (to substitute the greater than 0 sign)
    at which point I get k^2=-5.
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    Re: Is this the discriminate?

    The discriminant is positive for all k. Indeed, k^2 >= -5 holds for all k because k^2 is nonnegative. The fact that the equation k^2 = -5 has only complex roots says that it has no real roots. Therefore, (since the function k^2 + 5 is continuous,) its value is either always positive or always negative. In this case, it is obviously always positive.
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    Re: Is this the discriminate?

    Quote Originally Posted by Mukilab View Post
    Can you please explain how to get those real roots? The method that I learned from my book clearly isn't working. My method:
    I got the discriminate with no problem
    but then I equalled it to 0 (to substitute the greater than 0 sign)
    at which point I get k^2=-5.
    Consider the equation ax^2+bx+c=0,~a\ne 0~.
    The discriminate is \Delta=b^2-4ac~. It is called that because it discriminates between real and complex roots.

    The roots are \frac{-b\pm\sqrt{\Delta}}{2a}. So if \Delta>0 there are two real roots; if \Delta=0 there is one real root; and if \Delta<0 there are two complex roots.
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    Senior Member Mukilab's Avatar
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    Re: Is this the discriminate?

    Quote Originally Posted by emakarov View Post
    The discriminant is positive for all k. Indeed, k^2 >= -5 holds for all k because k^2 is nonnegative. The fact that the equation k^2 = -5 has only complex roots says that it has no real roots. Therefore, (since the function k^2 + 5 is continuous,) its value is either always positive or always negative. In this case, it is obviously always positive.
    Please elaborate, I don't quite understand what you're saying. Are you saying that my answer through solving the discriminant was right? Or are you saying simply my calculation was right but the answer to the discriminant is something completely different?

    If it is the latter, please can you explain how to find the right value for k (that is not a complex number)?
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    Re: Is this the discriminate?

    Quote Originally Posted by Mukilab View Post
    Please elaborate, I don't quite understand what you're saying. Are you saying that my answer through solving the discriminant was right? Or are you saying simply my calculation was right but the answer to the discriminant is something completely different?
    If it is the latter, please can you explain how to find the right value for k (that is not a complex number)?
    Reread all the replies.
    Yes we are saying that you method is completely off.
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    Senior Member Mukilab's Avatar
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    Re: Is this the discriminate?

    Quote Originally Posted by Plato View Post
    Consider the equation ax^2+bx+c=0,~a\ne 0~.
    The discriminate is \Delta=b^2-4ac~. It is called that because it discriminates between real and complex roots.

    The roots are \frac{-b\pm\sqrt{\Delta}}{2a}. So if \Delta>0 there are two real roots; if \Delta=0 there is one real root; and if \Delta<0 there are two complex roots.
    Thank you for the explanation however the question specifically said to 'prove that, for all real values of k, the roots of the equation are real and different'.

    So surely I have to assume the discriminate is >0? But If I do this, as I have shown above I find that k is greater than a complex number. How can that 'prove' that there are real roots, if I only get a complex number as an outcome?
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    Senior Member Mukilab's Avatar
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    Re: Is this the discriminate?

    Quote Originally Posted by Plato View Post
    Reread all the replies.
    Yes we are saying that you method is completely off.
    Can you please explain what I'm doing wrong with the method?

    I'm finding b^2-4ac to be k^2+5>0

    Where do I go from there? How do I 'prove' that the roots of the equation are real and different?
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    Re: Is this the discriminate?

    Quote Originally Posted by Mukilab View Post
    Thank you for the explanation however the question specifically said to 'prove that, for all real values of k, the roots of the equation are real and different'.
    So surely I have to assume the discriminate is >0? But If I do this, as I have shown above I find that k is greater than a complex number. How can that 'prove' that there are real roots, if I only get a complex number as an outcome?
    For every k, 4k^2+20>0. So that are two real roots for every k.

    If you still do not get it, you need a sit-down with a live tutor.
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    Re: Is this the discriminate?

    No, you have NOT shown that "that k is greater than a complex number"- for two reasons. First, by hypothesis, k is a real number so it makes no sense to say that k is "greater than an imaginary number" (which is what you really meant). Second, there is no way to define ">" to make the complex numbers an ordered field.

    What you have shown is that k^2> -5 which, because a real number, squared, is never negative, is true for any value of k.
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  12. #12
    Senior Member Mukilab's Avatar
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    Re: Is this the discriminate?

    Quote Originally Posted by Plato View Post
    For every k, 4k^2+20>0. So that are two real roots for every k.

    If you still do not get it, you need a sit-down with a live tutor.
    Thank you, I understand now and I'm sorry for taking so long to understand it, it seems very simple now.

    I just had the misconception that I had to do what I did with other equations such as (4k-20>0) which is to simplify to find k (i.e. getting 4k>20.... k>5). I never thought it could be so simple because it is literally just a two line answer.

    Once again, sorry for my incompetence. I guess I am making it overcomplicated.
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    Re: Is this the discriminate?

    Quote Originally Posted by Mukilab View Post
    Can you please explain what I'm doing wrong with the method?
    I'm finding b^2-4ac to be k^2+5>0
    Where do I go from there? How do I 'prove' that the roots of the equation are real and different?
    In x^2+2kx-5=0, a=1,~b=2k,~\&~c=-5.

    So \Delta=b^2-4ac=(2k)^2-4(1)(-5)=4k^2+20~!
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  14. #14
    Senior Member Mukilab's Avatar
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    Re: Is this the discriminate?

    Just as a last note of clarification.

    If I wanted to prove an equation had two real roots 'by finding the discriminant' all I would have to write down is:

    (It's discriminant)>0
    I.E.
    b^2-4ac>0 with the values of the equation substituted in.

    Is this correct?
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    Re: Is this the discriminate?

    Quote Originally Posted by Mukilab View Post
    Just as a last note of clarification.
    If I wanted to prove an equation had two real roots 'by finding the discriminant' all I would have to write down is:
    (It's discriminant)>0
    I.E.
    b^2-4ac>0 with the values of the equation substituted in.
    If \Delta>0 there are two real roots.
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