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What will be the difference of the roots of the equation $$ (x^2 - 10x - 29)^{-1} + (x^2 - 10x - 45)^{-1} = 2(x^2 - 10x - 69)^{-1} $$

I actually tried to solve it but it was too lengthy to calculate with the simple method I know. I want to solve it in less time coz someone told me that it could be solved in less time. Can someone help?

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    Hint: substitute $x^2-10x = y$ and solve for $y$ first. – 2017-01-08
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    Thanks a lot, I got y = 39, and the roots as -3 and 13, so the difference of the roots = 16. – 2017-01-08
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    @dxiv Thanks a lot, I got y = 39, and the roots as -3 and 13, so the difference of the roots = 16. – 2017-01-08

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$$x^2-10x-45=y$$ then

$$\frac{1}{y+16}+\frac{1}{y}=\frac{2}{y-24}$$

$$(2y+16)(y-24)=(y+16)2y \rightarrow y=-6 $$

Can you finish?

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    Thanks I got y = -6 and the difference of the roots to be 16 – 2017-01-08
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    The roots are -3 and 13 – 2017-01-08
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    You are very welcome! – 2017-01-08