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Solve for x: \(\displaystyle \frac{1}{x^{2} – 10x - 45} + \frac{1}{x^{2} +10x - 29} = \frac{2}{x^{2} – 10x - 69}\)

 Jul 9, 2020
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Let 

\(a = x^2 - 10x - 45\\ b = x^2 + 10x - 29\\ c = x^2 - 10x - 69\)

 

\(\dfrac1a + \dfrac1b = \dfrac2c\\ 2ab = c(a + b)\\ 2(x^2 - 10x - 45)(x^2 + 10x - 29) = 2(x^2 - 10x - 69)(x^2 - 37)\)

\((x^2 - 10x - 45)(x^2 + 10x - 29) = (x^2 - 10x - 69)(x^2 - 37)\)

 

Then expand and simplify.

 

\(5x^3 - 34x^2 - 265x - 624 = 0\)

 

Solving by numerical methods gives \(x\approx 12.055\).

 Jul 9, 2020

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