Rational Equation Real Roots
Find the real roots of
\[\frac{( x+ 1)(x - 3)}{5(x + 2)(x - 4)} + \frac{(x + 3)(x - 5)}{9(x + 4)(x - 6)} - \frac{2(x + 5)(x - 7)}{13(x + 6)(x - 8)} = \frac{92}{585}.\]Enter the real roots, separated by commas.
- 1
- 2
- 3
- +
- 4
- 5
- 6
- -
- 7
- 8
- 9
- $\frac{a}{b}$
- .
- 0
- =
- %
- $a^n$
- $a^{\circ}$
- $a_n$
- $\sqrt{}$
- $\sqrt[n]{}$
- $\pi$
- $\ln{}$
- $\log$
- $\theta$
- $\sin{}$
- $\cos{}$
- $\tan{}$
- $($
- $)$
- $[$
- $]$
- $\cap$
- $\cup$
- $,$
- $\infty$
Solution
Multiplying out each numerator and denominator, we get
\[\frac{x^2 - 2x - 3}{5(x^2 - 2x - 8)} + \frac{x^2 - 2x - 15}{9(x^2 - 2x - 24)} - \frac{2(x^2 - 2x - 35)}{13(x^2 - 2x - 48)} = \frac{92}{585}.\]We can write this as
\[\frac{(x^2 - 2x - 8) + 5}{5(x^2 - 2x - 8)} + \frac{(x^2 - 2x - 24) + 9}{9(x^2 - 2x - 24)} - \frac{2((x^2 - 2x - 48) + 13)}{13(x^2 - 2x - 48)} = \frac{92}{585}.\]Hence,
\[\frac{1}{5} + \frac{1}{x^2 - 2x - 8} + \frac{1}{9} + \frac{1}{x^2 - 2x - 24} - \frac{2}{13} - \frac{2}{x^2 - 2x - 48} = \frac{92}{585}.\]This simplifies to
\[\frac{1}{x^2 - 2x - 8} + \frac{1}{x^2 - 2x - 24} - \frac{2}{x^2 - 2x - 48} = 0.\]Let $ y = x^2 - 2x - 48 $. Then
\[\frac{1}{y + 40} + \frac{1}{y + 24} - \frac{2}{y} = 0.\]Multiplying everything by $ y(y + 24)(y + 40), $ we get
\[y(y + 24) + y(y + 40) - 2(y + 24)(y + 40) = 0.\]This simplifies to $ 64y + 1920 = 0, $ so $ y = -30 $. Then $ x^2 - 2x - 48 = -30, $ or $ x^2 - 2x - 18 = 0 $. By the quadratic formula, $ x = \boxed{1 \pm \sqrt{19}} $. (Since the denominators are nonzero for these values, we know that they are not extraneous.)
Freeflows