How do you solve a/(2a+1) - (2a^2+5)/ (2a^2-5a-3) =3/(a-3)?

May 26, 2015

Given $\frac{a}{2 a + 1} - \frac{2 {a}^{2} + 5}{2 {a}^{2} - 5 a - 3} = \frac{3}{a - 3}$

Noting that $2 {a}^{2} - 5 a - 3 = \left(2 a + 1\right) \left(a - 3\right)$
we can clear the denominators from the given equation by multiplying by $2 {a}^{2} - 5 a - 3$

$a \left(a - 3\right) - \left(2 {a}^{2} + 5\right) = 3 \left(2 a + 1\right)$

${a}^{2} - 3 a - 2 {a}^{2} - 5 = 6 a + 3$

${a}^{2} + 9 a + 8 = 0$

Factor the left side
$\left(a + 1\right) \left(a + 9\right) = 0$

State the solutions
$a = - 1$ or $a = - 9$