a/(a+3) = (2a)/(a - 3) - 1
To solve the given equation:
a/(a+3) = (2a)/(a - 3) - 1
First, let's simplify the equation:
Multiply both sides of the equation by (a+3)(a-3) to eliminate the fractions:
(a)(a-3) = (2a)(a+3) - (a+3)(a-3)
Simplify the equation further:
a(a-3) = 2a(a+3) - (a+3)(a-3)
Expand and simplify the equation:
a^2 - 3a = 2a^2 + 6a - (a^2 - 9)
Now, continue to solve the equation:
a^2 - 3a = 2a^2 + 6a - a^2 + 9
Rearrange the equation:
0 = 2a^2 + 6a - a^2 + 9 - a^2 + 3a
Combine like terms:
0 = 2a^2 - a^2 - a^2 + 6a + 3a + 9
0 = a^2 + 9a + 9
Now, the equation is quadratic. To solve it, we can use the quadratic formula:
a = (-b ± √(b^2 - 4ac)) / (2a)
In this case, the quadratic equation is:
a^2 + 9a + 9 = 0
a = [-(9) ± √((9)^2 - 4(1)(9))] / (2(1))
Simplify the equation further:
a = [-9 ± √(81 - 36)] / 2
a = [-9 ± √45] / 2
Therefore, the solutions to the equation are:
a = (-9 + √45) / 2
and
a = (-9 - √45) / 2