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CH-07-001 · Rational equation with two solutions

Solve 8x−9+4x+9=1\frac{8}{x-9}+\frac{4}{x+9}=1. Give exact answers and state the excluded values of x.

Solving Equations: Algebraic Fractions + Quadratic Formula Booklet p. 12 & p. 13
Method
1) Cross multiply over common denominator (x − h)(x + h)
2) Kill denominator and rearrange to ax² + bx + c = 0
3) Solve using Quadratic Formula
Worked Example (CH-07-001)
1)
Excluded values: x≠9, x≠−9x\ne 9,\ x\ne-9.
2)
Use common denominator (x−9)(x+9)(x-9)(x+9) and multiply both sides by it: 8(x+9)+4(x−9)=(x−9)(x+9)8(x+9)+4(x-9)=(x-9)(x+9).
3)
12x+36=x2−81⇒x2−12x−117=012x + 36=x^2-81\Rightarrow x^2 - 12x - 117=0
4)
Bracketed quadratic-formula substitution (p. 13): x=−(−12)±(−12)2−4(1)(−117)2(1)=12±6122x=\frac{-(-12)\pm\sqrt{(-12)^2-4(1)(-117)}}{2(1)}=\frac{12\pm\sqrt{612}}{2}.
5)
Check the excluded values in the cleared equation: at x=9x=9, left side is 144, right side is 0; at x=−9x=-9, left side is -72, right side is 0. Neither is a solution.
Ans = x=12±6122x=\frac{12\pm\sqrt{612}}{2}; excluded x = ±9.