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Thread: Unknown roots of two quadratics

  1. #1
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    Unknown roots of two quadratics

    Let $\displaystyle (a, c)$ and $\displaystyle (b, d)$ be the roots of the equation $\displaystyle x^2 + ax - b = 0$ and $\displaystyle x^2 + cx + d = 0$ respectively. Find all possible real values for $\displaystyle a, b, c, d$

    I'm not sure how to do this question and not sure how to approach it.

    Please help, BG
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  2. #2
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    Hello,

    In a quadratic equation $\displaystyle x^2+mx+n=0$, m is the sum of the roots, and n is the product of the roots.

    So from the second equation, we have $\displaystyle bd=d$ (1). And $\displaystyle c=b+d$ (2)
    And from the first one, we have $\displaystyle ac=-b$ (3) and $\displaystyle a+c=a$ (4)

    From (4), we get c=0
    Substituting in (3), we get b=0
    Substituting in (1), we get d=0

    Then, a can take any values you want...
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