### Instruction: Enter all the solutions to
\[ \sqrt{4x-3}+\frac{10}{\sqrt{4x-3}}=7,\]separated by commas.### Response: This is a quadratic equation in disguise, because if I let y = \sqrt{4x-3}, then I can rewrite the equation as y + 10/y = 7.
To solve for y, I need to get rid of the fraction, so I multiply both sides by y, and get y^2 + 10 = 7y.
Now I can bring everything to one side and factor, and get y^2 - 7y + 10 = 0.
This factors into (y-5)(y-2) = 0, so y = 5 or y = 2.
But y is not the final answer, I need to find x.
To do that, I use the original substitution, y = \sqrt{4x-3}, and square both sides, and get 4x-3 = y^2.
If y = 5, then 4x-3 = 25, and x = 7.
If y = 2, then 4x-3 = 4, and x = 7/4.
So the solutions are x = 7 and x = 7/4.