Question

Solve for x.
3x-1/x+3 +3= 4x/x-3

Answer 1

We start by combining the fraction on the left side and the fraction on the right side. We note that the lowest common denominator for both fractions is (x+3)(x-3). So we get,

3x-1/x+3 +3= (3x-1)(x-3)/(x+3)(x-3) +3(x+3)(x-3)/(x+3)(x-3)

4x/x-3 =4x(x+3)/(x+3)(x-3)

Now that both fractions have the same denominator, we can equate the numerators

(3x-1)(x-3) +3(x+3)(x-3) = 4x(x+3)
3x^2 - 10x + 3 + 3x^2 -9x = 4x^2 + 12x
6x^2 - 19x + 3= 4x^2 + 12x
2x^2 - 31x + 3= 0

Now, we solve the quadratic equation 2x^2 - 31x + 3= 0. Using the quadratic formula, we get

x = (31 ± sqrt(31^2 - 4*2*3)) / (2*2)
x = (31 ± sqrt(937)) / 4

The two possible values of x are (31 + sqrt(937))/4 and (31- sqrt(937))/4.