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Thread: discriminant

  1. #1
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    discriminant

    hi, i just want to check my answer against someone else's here, my textbook tells me a different answer to what i'm getting. i think the books wrong, could someone tell me if i'm right please:

    by considering the discrimination of the resulting quadratic, determine the number of points of intersection of the following line and curve:

    line $\displaystyle 3x - 2y = 5$ and curve $\displaystyle y = 3x^2 - 4x - 2$

    the book says two points of intersection and discriminant = 97
    but i say two points of intersection and discriminant = 241

    thanks
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  2. #2
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    Quote Originally Posted by mark View Post
    hi, i just want to check my answer against someone else's here, my textbook tells me a different answer to what i'm getting. i think the books wrong, could someone tell me if i'm right please:

    by considering the discrimination of the resulting quadratic, determine the number of points of intersection of the following line and curve:

    line $\displaystyle 3x - 2y = 5$ and curve $\displaystyle y = 3x^2 - 4x - 2$

    the book says two points of intersection and discriminant = 97
    but i say two points of intersection and discriminant = 241

    thanks
    Are you sure the answer given wasn't $\displaystyle \frac{97}{4}$?

    They will intersect where the functions are equal.

    $\displaystyle y = \frac{3}{2}x - \frac{5}{2}$

    $\displaystyle y = 3x^2 - 4x - 2$.


    Thus

    $\displaystyle \frac{3}{2}x - \frac{5}{2} = 3x^2 - 4x - 2$

    $\displaystyle 0 = 3x^2 - \frac{11}{2}x + \frac{1}{2}$.


    The discriminant

    $\displaystyle \Delta = \left(-\frac{11}{2}\right)^2 - 4\left(3\right)\left(\frac{1}{2}\right)$

    $\displaystyle = \frac{121}{4} - 6$

    $\displaystyle = \frac{97}{4}$.


    So the discriminant is $\displaystyle \frac{97}{4}$ so there will be two points of intersection.
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    no it just says 97, you've given a very complicatied way of figuring it out compared to the books method
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    Quote Originally Posted by mark View Post
    no it just says 97, you've given a very complicatied way of figuring it out compared to the books method
    I don't see what's so complicated...

    Let's try it a different way

    Substitute $\displaystyle y = 3x^2 - 4x - 2$ into the first equation.


    $\displaystyle 3x - 2(3x^2 - 4x - 2) = 5$

    $\displaystyle 3x - 6x^2 + 8x + 4 = 5$

    $\displaystyle 0 = 6x^2 - 11x + 1$


    Checking the discriminant

    $\displaystyle \Delta = (-11)^2 - 4(6)(1)$

    $\displaystyle = 121 - 24$

    $\displaystyle = 97$.


    So the answer is two solutions with $\displaystyle \Delta = 97$.



    Having said that, my first method was also correct, because notice that if you divide both sides of the equation

    $\displaystyle 0 = 6x^2 - 11x + 1$

    by 2, we get the same equation as before

    $\displaystyle 0 = 3x^2 - \frac{11}{2} + \frac{1}{2}$.
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  5. #5
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    you're right, the method i used was a bit ridiculous now i look at it, probably why i got it wrong. thanks though
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