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Math Help - sum and product of roots

  1. #1
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    sum and product of roots

    I have just been reading about the relationship between the roots of a quadractic and the coefficents.

    The book used identities to show if x , y are the roots of the equation aqz^2+bz+c=0, then -b/a=x+y and c/a=xy. Then it asked me to find the equation with roots whose sum and product were known.

    I was doing the questions but I was thinking surely something else needs to be true. Am I correct in saying in order to do this question, you need to know that the converse is true as well. I.E. If x+y =-b/a and xy=c/a, then x,y are the roots of the equation. This can be seen by solving the relations simultaneously and finding that the only 2 x,y satisfying the relations are the two roots of the quadractic equation.

    Thanks for reading and answering.
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  2. #2
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    Re: sum and product of roots

    Quote Originally Posted by Duke View Post
    The book used identities to show if x , y are the roots of the equation az^2+bz+c=0, then -b/a=x+y and c/a=xy. Then it asked me to find the equation with roots whose sum and product were known.
    The roots of az^2+bz+c=0 are the roots of z^2+\frac{b}{a}z+\frac{c}{a}=0 (of course a\ne 0).

    But by the factor theorem
    z^2+\frac{b}{a}z+\frac{c}{a}=(z-x)(z-y)=z^2-(x+y)z+(xy).
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  3. #3
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    Re: sum and product of roots

    Hello, Duke!

    Your use of x and y is confusing.
    I'll modify it to "standard" notation.


    \text{I have just been reading about the relationship}
    . . \text{between the roots of a quadractic and the coefficents.}

    \text{The book used identities to show:}
    . . \text{If }p, q\text{ are the roots of the equation }ax^2+bx+c\:=\:0,
    . . \text{ then: }\:p+q \,=\,\text{-}\tfrac{b}{a}\,\text{ and }\,pq \,=\,\tfrac{c}{a}


    \text{Then it asked me to find the equation with roots}
    . . \text{whose sum and product were known.}

    \text{I was doing the questions, but I was thinking:}
    . . \text}surely something else needs to be true.}

    \text{Am I correct in saying in order to do this question,}
    . . \text{you need to know that the converse is true as well?}

    \text{That is: \:if }p+q \,=\,\text{-}\tfrac{b}{a}\,\text{ and }\,pq\,=\,\tfrac{c}{a},
    . . \text{ then }p,q\text{ are the roots of the equation.}

    Good thinking!

    Yes, you are correct.

    Those roots/coefficients relations are usually given
    . . (or should be given) with an "if and only if".

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  4. #4
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    Re: sum and product of roots

    Thanks for both helpful replies.
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