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Math Help - How to solve for c, given the roots of the equation?

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    How to solve for c, given the roots of the equation?

    Hi all,

    How do I find the value of c for the question below?

    The roots of the equation x^2 + 6x + c = 0 are k and k-1. Find the value of c.
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    Re: How to solve for c, given the roots of the equation?

    Quote Originally Posted by bartholomew View Post
    Hi all,

    How do I find the value of c for the question below?

    The roots of the equation x^2 + 6x + c = 0 are k and k-1. Find the value of c.
    Solving the quadratic equation

    \displaystyle \begin{align*} x^2 + 6x + c &= 0 \\ x^2 + 6x + 3^2 - 3^2 + c &= 0 \\ (x + 3)^2 + c - 9 &= 0 \\ (x + 3)^2 &= 9 - c \\ x + 3 &= \pm \sqrt{9 - c} \\ x &= -3 \pm \sqrt{9 - c} \end{align*}

    So \displaystyle k = -3 + \sqrt{9-c} and \displaystyle k - 1 = -3 - \sqrt{9 - c}.

    Solve these for \displaystyle c
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    Re: How to solve for c, given the roots of the equation?

    Quote Originally Posted by Prove It View Post
    Solving the quadratic equation

    \displaystyle \begin{align*} x^2 + 6x + c &= 0 \\ x^2 + 6x + 3^2 - 3^2 + c &= 0 \\ (x + 3)^2 + c - 9 &= 0 \\ (x + 3)^2 &= 9 - c \\ x + 3 &= \pm \sqrt{9 - c} \\ x &= -3 \pm \sqrt{9 - c} \end{align*}

    So \displaystyle k = -3 + \sqrt{9-c} and \displaystyle k - 1 = -3 - \sqrt{9 - c}.

    Solve these for \displaystyle c
    Hi Prove It,

    Thanks for your reply. Am very confused with the solution. why do you have the  3^2 - 3^2

    Anyway, I solve for c and I come up with c being 21.25, can you confirm if that is correct? How do I check that the value of c that I come up with is correct?
    Last edited by bartholomew; November 16th 2011 at 12:28 AM. Reason: Additional text
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    Re: How to solve for c, given the roots of the equation?

    Hello, bartholomew!

    Another approach . . .


    \text{The roots of the equation }\,x^2 + 6x + c \,=\, 0\,\text{ are }k\text{ and }k-1.
    \text{Find the value of }c.

    Since x = k is a root of the equation,
    . . we have: . k^2 + 6k + c \:=\:0 .[1]

    Since x=k+1 is a root of the equation,
    . . we have: . (k-1)^2 + 6(k-1) + c \:=\:0 \quad\Rightarrow\quad k^2 + 4k + c - 5 \:=\:0 .[2]

    Subtract [1] - [2]: . 2k + 5 \:=\:0 \quad\Rightarrow\quad k \,=\,\text{-}\tfrac{5}{2}

    Substitute into [1]: . (\text{-}\tfrac{5}{2})^2 + 6(\text{-}\tfrac{5}{2}) + c \:=\:0 \quad\Rightarrow\quad \boxed{c \:=\:\frac{35}{4}}

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    Re: How to solve for c, given the roots of the equation?

    Quote Originally Posted by bartholomew View Post
    How do I find the value of c for the question below?
    The roots of the equation x^2 + 6x + c = 0 are k and k-1. Find the value of c.
    Here is a third way.
    The sum of the roots: k+(k-1)=-6.

    The product of the roots: k(k-1)=c.
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