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Math Help - Quadratics #3

  1. #1
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    Exclamation Quadratics #3

    Hi

    For the question:

    Show that the equation (x^2 + 5x)/(7x + 9) = (1 - k)/(1 + k) has real roots for all values of k.

    I know that for real roots, the discriminant has to be greater or equal than 0. I found out the discriminant, which is x^2 + kx^2 - 2x + 12 xk - 9k + 9 greater than or equal to 0. How do I make it into a quadratic equation?

    Thanx
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  2. #2
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    Hello,
    Quote Originally Posted by xwrathbringerx View Post
    Hi

    For the question:

    Show that the equation (x^2 + 5x)/(7x + 9) = (1 - k)/(1 + k) has real roots for all values of k.

    I know that for real roots, the discriminant has to be greater or equal than 0. I found out the discriminant, which is x^2 + kx^2 - 2x + 12 xk - 9k + 9 greater than or equal to 0. How do I make it into a quadratic equation?

    Thanx
    No, what you got is just the equation from which you want to find the discriminant.

    It's x+kx-2x+12kx+9k-9=0

    which can be grouped this way :
    (1+k)x+x(12k-2)+(9k-9)=0

    you know that the discriminant of ax+bx+c is b-4ac.
    Here, a=1+k, b=12k-2, c=9k-9.


    note that 7x+9 \neq 0, that is to say x \neq -\frac 97


    and you'll have to state the case for k+1 \neq 0, that is to say k \neq -1
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  3. #3
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    Hmm I keep on getting 27k^2 - 12k + 10 as the discriminant but when I try to find k, there are no real solutions. What does that mean?
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    Quote Originally Posted by xwrathbringerx View Post
    Hmm I keep on getting 4(27k^2 - 12k + 10) as the discriminant but when I try to find k, there are no real solutions. What does that mean?
    Good work, the discriminant is correct =)

    Now, you want the discriminant to be positive. Once again, find the determinant of the quadratic 27k-12k+10 :
    - if it has no real root (that is to say the discriminant is < 0), then it is always positive, which is what you want
    - if the discriminant is = 0, then it is positive (as a square)
    - if the discriminant is > 0, it can be either positive or negative (in general)
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