# Substitutions in a square root equation

1. Feb 17, 2009

### tom92373

I recently came across the equation: 7sqrt(3x+10) - 4sqrt(9x+19) = sqrt((3x-14)(2x-1))
The solutions are: 5, -2, -1/3. I can only solve the equation by squaring both sides which leads to huge coefficients, yet it does give the correct solutions. I was trying to find a way make a clever substitution or series of substitutions that will "clean up" the equation to something more manageable. I made one substitution that may have done some good: sqrt(6)x - 31/2 = (sqrt(905)/2)sec(theta) It simplified the equation to: 7sqrt(3x+10) - 4sqrt(9x+19) = (sqrt(905)/2)tan(theta) I don't know if this does any good... anyway, I would like to solve this equation with a substitution not leading to large coefficients. Tanks!