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Math Help - Location of roots of two quadratic w.r.t. to third quadratic

  1. #1
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    Location of roots of two quadratic w.r.t. to third quadratic

    If a,b,c \in R such that abc \neq0 If x_1 is a root of a^2x^2+bx+c=0, x_2 is a root of a^2x^2-bx-c=0 and x_1 > x_2 >0 then the equation a^2x^2+2bx+2c=0 has roots x_3 .

    Prove that x_3 lies between x_1 \& x_2

    Let f(x) = a^2x^2+2bx+2c=0

    $\Rightarrow f(x_1)=a^2x_1^2+2bx_1+2c=-a^2x_1^2

    \Rightarrow f(x_2) = a^2x_2^2+2bx_2+2c=3a^2x_2^2

    \Rightarrow f(x_1)(x_2) = (3a^2x_2^2)(-a^2x_1^2) <0


    \RightarrowThus one root of a^2x^2+2bx+2c=0 will lie between x_1 \& x_2

    Please provide explanation on the last statement of this answer how it derived ...Thanks..
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  2. #2
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    Re: Location of roots of two quadratic w.r.t. to third quadratic

    Hey sachinrajsharma.

    What is the definition of f(x1)(x2)? Is it in terms of f(x1) and g(x2)?
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  3. #3
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    Re: Location of roots of two quadratic w.r.t. to third quadratic

    f(x_{1})f(x_{2})<0 implies that f(x_{1}) \text{  and  } f(x_{2}) are opposite in sign.
    Since f(x) is continuous, it must be zero at some point between x_{1} and x_{2}.
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