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Indirect Proof

  1. Sep 25, 2006 #1
    Heres the quesion

    Prove that the line whose equation is y=2x-1 does not intersect the curve with equation y=x^4 + 3x^2 +2x.

    We are suppose to solve this using indirect proof, thus assuming the equations do intersect, and proving that wrong.

    i let the y's equal each other, but that isnt getting me anywhere

    where should i start.

  2. jcsd
  3. Sep 25, 2006 #2


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    Homework Helper

    Of course that is getting you somewhere. Ever thought of substituting x^2 = t ?
  4. Sep 25, 2006 #3
    Let [itex]y_1(x) = 2x-1[/itex] and [itex]y_2(x) = x^4 + 3x^2 + 2x[/itex]. Look at [itex]f(x) = y_2(x) - y_1(x) = x^4 + 3x^2 + 1[/itex].

    If [itex]y_1[/itex] and [itex]y_2[/itex] intersect at [itex]x_0 \in \mathbb{R}[/itex] then [itex]f(x_0) = 0[/itex]. Can you get the contradiction (what do you know about [itex]x^2, \, x^4[/itex] when [itex]x \in \mathbb{R}[/itex]?)?
  5. Sep 25, 2006 #4

    umm yea i got the roots, x=1 or x=1 or x=-1

    but what does that mean
  6. Sep 25, 2006 #5


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    How did you get these roots? Let's start again. Intersection means setting x^4+3x^2+2x = 2x - 1, which implies x^4+3x^2+1=0. Now, as said, substitute x^2 = t, and solve the quadratic equation. Both solutions of this equation t1 and t2 are negative. So, substituting back to x^2 = t means that there is no real solution for the equation x^4+3x^2+1=0, i.e. y1 = x^4+3x^2+2x and y2 = 2x - 1 don't intersect.
  7. Sep 25, 2006 #6
    thank you so much

    i just made a simple sign error in factoring t^2 + t +1

    which gave me wrong roots

    thanks again
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