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Solving Equations with Algebraic Fractions

Solve \frac{5}{n} + \frac{2}{n^2-9} = \frac{6}{n-3}

  1. Common denominator left side (n) \cdot (n^2-9)
    1.  \frac{5}{n} \cdot \frac{n^2-9}{n^2-9} + \frac{2}{n^2-9} - \frac{n}{n} = \frac{6}{n-3}
    2. \frac{5(n^2-9)+2n}{n(n^2-9)}=\frac{6}{n-3}
  2. Move everything to the left side
    1. \frac {5(n^2-9)+2n} {n(n^2-9)}- \frac {6} {n-3}=0
  3. Common denominator left side (n)(n^2-9)
    1. (n^2-9) -> difference of squares
    2. n^2-9 =(n+3)(n-3)
    3. \frac{5(n^2-9)+2n}{n(n^2-9)}-\frac{6}{n-3} \cdot \frac{n(n+3)}{n(n+3)}=0
  4. Simplify and solve
    1. \frac{5(n^2-9)+2n-6n(n+3)}{n(n^2-9}=0
    2. 5n^2-45+2n-6n^2-18n=0
    3. -2n^2-16n-45=0
    4. n=\frac{-b\pm \sqrt{b^2-4ac}}{2a}
    5. =\frac{16\pm \sqrt{16^2-4(-1)(-45)}}{2(-1)}
    6. =\frac{16 \pm \sqrt{76}}{02}
    7. \sqrt{76}=\sqrt{4 \cdot 19}=2 \sqrt{19}
    8. \frac{16\pm 2\sqrt{19}}{-2}
    9. -8\pm \sqrt{19}