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Math Help - Polynomial Equations help

  1. #1
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    Polynomial Equations help

    Find the sum of the solutions of the equation

    (3x - 8/x)^2 - 7(3x - 8/x)^2 +10 = 0
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  2. #2
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    Hi there sri340

    Notice there are some like terms here

     \left( 3x-\frac{8}{x}\right)^2-7\left( 3x-\frac{8}{x}\right)^2+10 = 0

    -6\left( 3x-\frac{8}{x}\right)^2+10 = 0

    6\left( 3x-\frac{8}{x}\right)^2 = 10

    Now expand the brackets and apply the quadratic fomula to solve for x
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  3. #3
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    Hello, sri340!

    I suspect there is a typo . . .


    Find the sum of the solutions of the equation: . \left(3x - \frac{8}{x}\right)^2 - 7\left(3x - \frac{8}{x}\right) +10 \:=\: 0

    Expand: . 9x^2 - 48 + \frac{64}{x^2} - 21x + \frac{56}{x} + 10 \:=\:0 \quad\Rightarrow\quad 9x^2 -21x - 38 + \frac{56}{x} + \frac{64}{x^2} \:=\:0

    Multiply by x^2\!:\quad 9x^4 - 21x^3 - 38x^2 + 56x + 64 \:=\:0

    Divide by 9: . x^4 - \frac{7}{3}x^3 - \frac{38}{9}x^2 + \frac{56}{9}x + \frac{64}{9} \:=\:0

    The sum of the roots is the negative of the x^3 coefficient: . \frac{7}{3}

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  4. #4
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    Allow your equation to read a^2-7a+10=0.
    Then (a-5)(a-2)=0.
    a=5 or 2.

    "a" = 3x-8/x, so 5x=3x^2-8 and 2x=3x^2-8.
    Hence, 3x^2-5x-8=0 and 3x^2-2x-8=0
    have solutions (3x-8)(x+1)=0 and (3x+4)(x-2)=0.

    x= 8/3, -1, -4/3 and 2.

    These sum to 7/3.
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