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Thread: Function

  1. #1
    Junior Member
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    Post Function

    Hi there, I have a problem that I can not seem to figure out. It is asking on the condition that $\displaystyle a >0$ to prove that the quadratic expression $\displaystyle ax^2 +bx +c$ is positve for all real values of $\displaystyle x$ when $\displaystyle b^2 < 4ac$. then it saids based on that to find the range of values of p for which the quadratic function of $\displaystyle x $

    $\displaystyle
    f(x) \equiv 4x^2 + 4px - (3p^2 +4p -3)
    $

    is positive for all real values of $\displaystyle x $ illustrate your result by making sketch graphs of $\displaystyle f(x) $ for each of the cases $\displaystyle p=0$ and $\displaystyle p=1$

    Can some one out there help me to figure out this problem/
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  2. #2
    Super Member Bacterius's Avatar
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    First question : First, you know that $\displaystyle a> 0$. That means that if you sketch the graph for any permitted value of $\displaystyle p$, it will be a curved parabola open to the top. Therefore, there are only two possible situations : either it never touches the x-axis, therefore there are no real solutions and the equation is always positive, either it cuts the x-axis and there are solutions. But the equation cannot be all negative. Now, if $\displaystyle b^2 < 4ac$, that means that $\displaystyle b^2 - 4ac < 0$. Basically, it means that the discriminant is negative, thus the equation admits no real solution and therefore is positive for all real $\displaystyle x$ (you got to rewrite it properly though, I gave you the idea).

    Second question : you need to find all values of $\displaystyle p$ for which the given quadratic has only positive values (that is, has no real solution). Think of your equation like this :

    $\displaystyle ax^2 + bx + c$

    Where :

    $\displaystyle a = 4$
    $\displaystyle b = 4p$
    $\displaystyle c = -(3p^2 + 4p - 3)$

    Get the discriminant of this equation :

    $\displaystyle \Delta = b^2 - 4ac$

    By substituting your values :

    $\displaystyle \Delta = (4p)^2 - 4 \times 4 \times [-(3p^2 + 4p - 3)]$

    Now, what is the characteristic feature of a quadratic equation with no real solution (i.e. all values are positive, in our case) ? The discriminant is negative. Therefore you have :

    $\displaystyle \Delta < 0$

    $\displaystyle (4p)^2 - 4 \times 4 \times [-(3p^2 + 4p - 3)] < 0$

    $\displaystyle 16p^2 + 16 \times (3p^2 + 4p - 3) < 0$

    $\displaystyle 16p^2 + 48p^2 + 64p - 48 < 0$

    $\displaystyle 64p^2 + 64p - 48 < 0$

    Solve for $\displaystyle p$ and you will then have the range of $\displaystyle p$ when the quadratic equation has no real solution (always positive).

    Does it help ? Ask if you still need help.

    PS : why did you put a congruent sign ($\displaystyle \equiv$) on your function ?
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  3. #3
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    Quote Originally Posted by Bacterius View Post
    First question : First, you know that $\displaystyle a> 0$. That means that if you sketch the graph for any permitted value of $\displaystyle p$, it will be a curved parabola open to the top. Therefore, there are only two possible situations : either it never touches the x-axis, therefore there are no real solutions and the equation is always positive, either it cuts the x-axis and there are solutions. But the equation cannot be all negative. Now, if $\displaystyle b^2 < 4ac$, that means that $\displaystyle b^2 - 4ac < 0$. Basically, it means that the discriminant is negative, thus the equation admits no real solution and therefore is positive for all real $\displaystyle x$ (you got to rewrite it properly though, I gave you the idea).

    Second question : you need to find all values of $\displaystyle p$ for which the given quadratic has only positive values (that is, has no real solution). Think of your equation like this :

    $\displaystyle ax^2 + bx + c$

    Where :

    $\displaystyle a = 4$
    $\displaystyle b = 4p$
    $\displaystyle c = -(3p^2 + 4p - 3)$

    Get the discriminant of this equation :

    $\displaystyle \Delta = b^2 - 4ac$

    By substituting your values :

    $\displaystyle \Delta = (4p)^2 - 4 \times 4 \times [-(3p^2 + 4p - 3)]$

    Now, what is the characteristic feature of a quadratic equation with no real solution (i.e. all values are positive, in our case) ? The discriminant is negative. Therefore you have :

    $\displaystyle \Delta < 0$

    $\displaystyle (4p)^2 - 4 \times 4 \times [-(3p^2 + 4p - 3)] < 0$

    $\displaystyle 16p^2 + 16 \times (3p^2 + 4p - 3) < 0$

    $\displaystyle 16p^2 + 48p^2 + 64p - 48 < 0$

    $\displaystyle 64p^2 + 64p - 48 < 0$

    Solve for $\displaystyle p$ and you will then have the range of $\displaystyle p$ when the quadratic equation has no real solution (always positive).

    Does it help ? Ask if you still need help.

    PS : why did you put a congruent sign ($\displaystyle \equiv$) on your function ?
    I think that is how they have it in the book, I am guesing that is what it is.
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  4. #4
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    Post

    Quote Originally Posted by scrible View Post
    I think that is how they have it in the book, I am guesing that is what it is.
    Thank you very much. I really appreciate it
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  5. #5
    Super Member Bacterius's Avatar
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    Quote Originally Posted by scrible View Post
    Thank you very much. I really appreciate it
    No problem, have a nice day !
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