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Thread: Range of a specific value in quadratic equation to have 3 real roots

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    Range of a specific value in quadratic equation to have 3 real roots

    Can anyone help me with the this?



    Shenaya Perera
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    Re: Range of a specific value in quadratic equation to have 3 real roots

    Hey pererashenaya7.

    It means you have to find a value of k where you can get three solutions for x to satisfy f(x) = 0 which you will have to define.
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    Re: Range of a specific value in quadratic equation to have 3 real roots

    Quote Originally Posted by chiro View Post
    Hey pererashenaya7.

    It means you have to find a value of k where you can get three solutions for x to satisfy f(x) = 0 which you will have to define.
    Yes...I know that...:-)

    For example if the discriminant of a quadratic equation is greater than or equal to 0 it has 2 real roots and if it is equal to zero it has one real root.
    But in here we have to find the range of k it has three real roots.But how? And that's exactly where I've been stuck... :-)

    Shenaya Perera
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    Re: Range of a specific value in quadratic equation to have 3 real roots

    Let y= tan(x). Then the equation is y^2- y- k= 0. That has two distinct real roots if and only if (-1)^2- 4(1)(-k)= 4k+ 1\ge 0. Now, what about tan(x)? For a given value of y, how many roots does tan(x)= k have?
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    Re: Range of a specific value in quadratic equation to have 3 real roots

    Quote Originally Posted by HallsofIvy View Post
    Let y= tan(x). Then the equation is y^2- y- k= 0. That has two distinct real roots if and only if (-1)^2- 4(1)(-k)= 4k+ 1\ge 0. Now, what about tan(x)? For a given value of y, how many roots does tan(x)= k have?
    Two real roots...

    But the question asks for a range of k which the equation has three real roots??.That's where I've been stuck..

    Shenaya Perera
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    Re: Range of a specific value in quadratic equation to have 3 real roots

    Maybe I'm missing something in the question (see attached graph of $y=\tan^2{x}-\tan{x}$) ...

    For $k < -\dfrac{1}{4}$, $\tan^2{x}-\tan{x} = k$ has no real solutions.

    For $k \ge -\dfrac{1}{4}$, $\tan^2{x}-\tan{x}=k$ has an infinite number of real solutions.


    If we're considering only a single period of the function ...

    $k < -\dfrac{1}{4}$ yields no solutions, $k = -\dfrac{1}{4}$ yields a single solution, and $k > - \dfrac{1}{4}$ yields only two solutions.

    Sorry, but I'm not seeing how the equation yields three real solutions unless there is a restricted interval for $x$ that was unstated in the original problem ... someone please enlighten me.
    Attached Thumbnails Attached Thumbnails Range of a specific value in quadratic equation to have 3 real roots-img_1521.png  
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    Re: Range of a specific value in quadratic equation to have 3 real roots

    Quote Originally Posted by skeeter View Post
    Maybe I'm missing something in the question (see attached graph of $y=\tan^2{x}-\tan{x}$) ...

    For $k < -\dfrac{1}{4}$, $\tan^2{x}-\tan{x} = k$ has no real solutions.

    For $k \ge -\dfrac{1}{4}$, $\tan^2{x}-\tan{x}=k$ has an infinite number of real solutions.


    If we're considering only a single period of the function ...

    $k < -\dfrac{1}{4}$ yields no solutions, $k = -\dfrac{1}{4}$ yields a single solution, and $k > - \dfrac{1}{4}$ yields only two solutions.

    Sorry, but I'm not seeing how the equation yields three real solutions unless there is a restricted interval for $x$ that was unstated in the original problem ... someone please enlighten me.
    No you aren't missing anything...I've posted the exact question..
    I'm also stuck on the same case,I also don't see a way to find a range of k which this equation has 3 real roots..

    Shenaya Perera
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    Re: Range of a specific value in quadratic equation to have 3 real roots

    Quote Originally Posted by pererashenaya7 View Post
    No you aren't missing anything...I've posted the exact question..
    I'm also stuck on the same case,I also don't see a way to find a range of k which this equation has 3 real roots..

    Shenaya Perera
    ... then I advise you research the origin of this question or ask whoever assigned it for clarification.
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