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Thread: Sum & Product of the Roots of a Quadratic Equation

  1. #1
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    Sum & Product of the Roots of a Quadratic Equation

    Having a bit of trouble with this one. Can anyone help?

    Many thanks.

    Q.: (i)
    The roots of the equation $\displaystyle 2x^2+6x+3=0$ are $\displaystyle \alpha$ & $\displaystyle \beta$. Show that $\displaystyle \alpha^2+\beta^2=6$. (ii) The roots of $\displaystyle 2x^2+px+q=0$ are $\displaystyle 2\alpha+\beta$ & $\displaystyle \alpha+2\beta$. Find the value of p & the value of q.

    Attempt: (i) a = 2, b = 6, c = 3

    $\displaystyle \alpha+\beta=\frac{-b}{a}$ => $\displaystyle \frac{-6}{2}$ => -3

    $\displaystyle \alpha\beta=\frac{c}{a}$ => $\displaystyle \frac{3}{2}$

    Sum of roots: $\displaystyle \alpha^2+\beta^2$ => $\displaystyle (\alpha+\beta)^2-2\alpha\beta$ => $\displaystyle (-3)^2-2(\frac{3}{2})$ => $\displaystyle 9-\frac{6}{2}$ => 9 - 3 => 6

    (ii) a = 2, b = p, c = q

    Sum of roots: $\displaystyle 2\alpha+\beta+\alpha+2\beta$ => $\displaystyle 3\alpha+3\beta$ => $\displaystyle 3(\alpha+\beta)$ => $\displaystyle \frac{-b}{a}$ => $\displaystyle \frac{-p}{2}$ => $\displaystyle 3(\alpha+\beta)$ => $\displaystyle 3(\frac{-p}{2})$ => $\displaystyle \frac{-3p}{2}$

    Product of roots: $\displaystyle (2\alpha+\beta)(\alpha+2\beta)$ => $\displaystyle 2\alpha^2+5\alpha\beta+2\beta^2$ => $\displaystyle 2(\alpha^2+\beta^2)+5\alpha\beta$ => $\displaystyle 2((\alpha+\beta)^2-2\alpha\beta)+5\alpha\beta$ => $\displaystyle 2(\alpha+\beta)^2+\alpha\beta$ => $\displaystyle 2(\frac{-p}{2})^2+(\frac{c}{a})$ => $\displaystyle \frac{2p^2}{4}+\frac{q}{2}$ => $\displaystyle \frac{p^2}{2}+\frac{q}{2}$ => $\displaystyle \frac{p^2+q}{2}$

    $\displaystyle x^2-$(sum of roots)x + (product of roots) => $\displaystyle x^2-(\frac{-3p}{2})x+\frac{p^2+q}{2}$ => $\displaystyle 2x^2+3px+p^2+q$...

    Ans.: (From text book): p = 18, q = 39
    Last edited by GrigOrig99; Jun 20th 2012 at 10:14 AM.
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  2. #2
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    Re: Sum & Product of the Roots of a Quadratic Equation

    It seems that more informations are needed. Since if $\displaystyle \alpha=0,\beta=1$ then the roots will be $\displaystyle 1,2$. And hence $\displaystyle p=-6,q=4$.
    Also, if
    $\displaystyle \alpha=1,\beta=-1$ then the roots will be $\displaystyle 1,5$. And hence $\displaystyle p=0,q=-2$.
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  3. #3
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    Re: Sum & Product of the Roots of a Quadratic Equation

    I've made the changes to the question I think you're referring to. There was an earlier piece to the question, but I had already solved it and wasn't sure it was relevant.
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  4. #4
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    Re: Sum & Product of the Roots of a Quadratic Equation

    Yes, you do have to use the $\displaystyle \alpha$ and $\displaystyle \beta$ from part a).

    Sum of roots:
    $\displaystyle 3 \alpha + 3 \beta = -\frac{p}{2}$. From part a), $\displaystyle \alpha + \beta = -3$, substituting yields $\displaystyle -9 = -\frac{p}{2} \Rightarrow p = 18$

    Product of roots:
    $\displaystyle 2 \alpha^2 + 5 \alpha \beta + 2 \beta^2 = \frac{q}{2} \Rightarrow 2(\alpha^2 + \beta^2) + 5 \alpha \beta = \frac{q}{2}$. From part a), $\displaystyle \alpha^2 + \beta^2 = 6$, and $\displaystyle \alpha \beta = \frac{3}{2}$. Therefore,

    $\displaystyle 2(6) + 5(\frac{3}{2}) = \frac{q}{2} \Rightarrow \frac{39}{2} = \frac{q}{2} \Rightarrow q = 39$.
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  5. #5
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    Re: Sum & Product of the Roots of a Quadratic Equation

    So, from the first part you have
    $\displaystyle \alpha^2+\beta^2=6$ and also $\displaystyle \alpha+\beta=-3$.
    We also need $\displaystyle \alpha\beta$ which is from the first equation
    $\displaystyle \alpha\beta=\frac{3}{2}.$
    Now,
    $\displaystyle p=-((\alpha+2\beta)+(2\alpha+\beta))=-6(\alpha+\beta)=-6*-3=18.$

    $\displaystyle q=(2\alpha+\beta)(\alpha+2\beta)=4(\alpha^2+\beta^ 2)+10\alpha\beta=4*6+10\frac{3}{2}=39$
    Last edited by Kmath; Jun 20th 2012 at 12:24 PM.
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