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Math Help - Algebraic Inequalities

  1. #1
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    Algebraic Inequalities

    Hi, I've just learnt about roots of algebraic equations and have to solve this problem:

    Find the range of values of m for which the line y=mx cuts the curve y=5x-4-x˛ at two distinct points.

    I managed to get x˛ - (5 - m)x + 4 = 0 and I think i'm supposed to apply b˛-4ac?
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by erika View Post
    Hi, I've just learnt about roots of algebraic equations and have to solve this problem:

    Find the range of values of m for which the line y=mx cuts the curve y=5x-4-x˛ at two distinct points.

    I managed to get x˛ - (5 - m)x + 4 = 0 and I think i'm supposed to apply b˛-4ac?
    not just b^2 - 4ac, the whole quadratic formula: x = [-b +/- sqrt(b^2 - 4ac)]/2a

    try to complete the problem using that. if you run into difficulties, get back to us
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  3. #3
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    I don't get it, I can't seem to simplify it using the formula...
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  4. #4
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by erika View Post
    I don't get it, I can't seem to simplify it using the formula...
    we want

    mx = 5x - 4 - x^2
    => x^2 + (m - 5)x + 4 = 0

    by the quadratic formula:

    => x = [-(m - 5) +/- sqrt{(m - 5)^2 - 4(1)(4)}]/2
    => x = [-(m - 5) +/- sqrt{m^2 - 10m + 25 - 16}]/2
    => x = [-(m - 5) +/- sqrt{m^2 - 10m + 9}]/2
    => x = [-(m - 5) +/- sqrt{(m - 9)(m - 1)}]/2

    we will have real roots as long as what is under the squareroot is nonnegative, that is greater than or equal to zero. so the range we want is:

    (m - 9)(m - 1) >= 0
    => m >= 9 or m <= 1 .......i had to check the ranges on a number line to find what values work in these ranges

    so the range of m we want is:

    (-infinity , 1] U [9 , infinity)

    still don't get it?
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  5. #5
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    I get it now, thank you very much =)
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