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Math Help - need to solve this problem

  1. #1
    Member grgrsanjay's Avatar
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    Wink need to solve this problem

    a<b<c<d
    f(x)=(x-a)(x-c)-3(x-b)(x-d)
    show that the roots of this equation are distinct.
    Last edited by grgrsanjay; August 4th 2010 at 05:09 AM.
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  2. #2
    MHF Contributor

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    The simplest way to do this is to look at the values of f(a), f(b), f(c), f(d)

    f(a)= -3(a- b)(a- d). Since a< b and a< d, both a- b and a- d are less than 0 so f(a)< 0.

    f(b)= (b- a)(b- c). a< b but b< c so b- a is positive, b- c is negative so f(b)< 0

    f(c)= -2(c- b)(c- d). b< c but c< d so c- b is positive and c-d is negative so f(c)< 0

    f(d)= (d- a)(d- b). a< d and b< d so both d- a and d- b are positive. f(d)> 0 so there is a root between c and d.

    Multiplying out, f(d)= x^2- (a+ c)x+ ac- 3x^2+ 3(b+d)x- 3bd= -2x^2- (a+ c- b- d)x+ ac- 3bd

    Since the leading coefficient is negative, for sufficiently large x, f(x) will be negative. There must be a second root larger than d.
    Last edited by HallsofIvy; August 10th 2010 at 02:41 AM.
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  3. #3
    MHF Contributor
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    Alternatively,

    since f(c)<0 and f(d)>0, the roots of the quadratic are real and distinct since the graph crosses the x-axis.

    The roots are non-distinct in the case of a double-root whereby the graph does not cross the x-axis.
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  4. #4
    Member grgrsanjay's Avatar
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    what if the question is   a<b<c<d
    f(x)=(x-a)(x-c)+(x-b)(x-d)
    does it have two distinct roots??
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