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  1. #1
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    straight line and function

    if a , b , c are odd numbers prove that the straight line ax+by+c=0 never

    intersect the function f(x)=x^2 in apoint its x-coordinate is rational number.
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    MHF Contributor ebaines's Avatar
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    Re: straight line and function

    To find the intersection point(s) substitute y = x^2 into the equation of the line ax+by+c = 0, to get ax + bx^2+c = 0. Now apply the quadratic formula to solve for values of x. Recall that in general the quadratic equation gives two roots (due to the "plus or minus" in front of the square root), unless the value of the square root is zero. So - given that the values a, b, and c are all odd, can that square root = 0?
    Last edited by ebaines; May 7th 2013 at 06:09 AM.
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    Re: straight line and function

    thank you ebaines, but it still not clear to me
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    MHF Contributor ebaines's Avatar
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    Re: straight line and function

    Quote Originally Posted by abualabed View Post
    thank you ebaines, but it still not clear to me
    Do you know the quadratic formula for finding the roots of a quadratic equation?
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    Re: straight line and function

    Shameful to ask me this question , ebaines
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    Re: straight line and function

    Quote Originally Posted by abualabed View Post
    Shameful to ask me this question , ebaines
    But please answer anyway. Saying what that has to do with your question.

    What are the roots of bx^2+ax+c=0~?
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    Re: straight line and function

    ok
    Attached Thumbnails Attached Thumbnails straight line and function-111111111111111111.jpg  
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    Re: straight line and function

    Quote Originally Posted by abualabed View Post
    ok
    If K\in\mathbb{Z}^+, then do you know under what conditions \sqrt{K} is rational?
    What does that have to do with the OP?
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    MHF Contributor ebaines's Avatar
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    Re: straight line and function

    Quote Originally Posted by abualabed View Post
    ok
    So if  \sqrt {a^2-4bc} = 0 there would be only one root for bx^2+ax+c=0 right? But if a, b, and c are all odd, is this possible? If not, then there must be either two real or two complex intersection points between the line and the parabola, not just one.
    Last edited by ebaines; May 7th 2013 at 09:27 AM.
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    Re: straight line and function

    Quote Originally Posted by ebaines View Post
    So if  \sqrt {a^2-4bc} = 0 there would be only one root for bx^2+ax+c=0 right? But if a, b, and c are all odd, is this possible? If not, then there must be either two real or two complex intersection points between the line and the parabola, not just one.
    @ebaines, Frankly, I don't understand your point.
    If a=3,~b=1,~\&~c=1 the line intersects in two point.
    But neither of the points has a rational x-coordinate. Why?

    Have a look at this,
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    MHF Contributor ebaines's Avatar
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    Re: straight line and function

    Quote Originally Posted by Plato View Post
    @ebaines, Frankly, I don't understand your point.
    If a=3,~b=1,~\&~c=1 the line intersects in two point.
    I interpret the OP's problem statement as proving that if a, b, and c are all odd then the line and parabola cannot touch at one point only. His phrase was: "straight line ax+by+c=0 never intersect the function f(x)=x^2 in apoint " I believe by "intersect ... in a point" he means the two functions touching at one point only, not two.

    Example: if a=6, b=3 and c=3 the two lines meet at only one point, at (-1,1). This is possible because a, b. and c are not all odd. But if a= 5, b=3, c = 1 they intersect at two points, and if a=5, b=3, c=3 they don't intersect at all.
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    Re: straight line and function

    Quote Originally Posted by ebaines View Post
    I interpret the OP's problem statement as proving that if a, b, and c are all odd then the line and parabola cannot touch at one point only. His phrase was: "straight line ax+by+c=0 never intersect the function f(x)=x^2 in apoint " I believe by "intersect ... in a point" he means the two functions touching at one point only, not two.

    Example: if a=6, b=3 and c=3 the two lines meet at only one point, at (-1,1). This is possible because a, b. and c are not all odd. But if a= 5, b=3, c = 1 they intersect at two points, and if a=5, b=3, c=3 they don't intersect at all.
    I could see reading it that way if the word tangent had been used. Otherwise, I disagree with that reading.

    I realize that this is a translated question. So who really knows.
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    Re: straight line and function

    hello Plato, ebaines

    I mean that" the x-coordinate of any intersection point never be rational number"
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    Re: straight line and function

    Quote Originally Posted by abualabed View Post
    hello Plato, ebaines
    I mean that" the x-coordinate of any intersection point never be rational number"
    That is exactly how I read the question and my replies reflect that reading.
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    MHF Contributor ebaines's Avatar
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    Re: straight line and function

    OK, thanks for clarifying. What you need to show is that for  \sqrt{a^2-4bc} to be rational at least one of the coefficients must be even. Consider  a^2-4bc = d^2, where d is an integer. Rearrange:  a^2-d^2 = 4bc, so if 'a' is odd then 'd' must also be odd since the right hand side is even. It's not too difficult to show that the difference of two odd numbers squared must be a mutiple of 8; therefore either 'b' or 'c' must be even.
    Last edited by ebaines; May 7th 2013 at 11:53 AM.
    Thanks from abualabed
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