3sqrt(5x-6) + 2sqrt(7-x) = 13

To solve the equation 3√(5x-6) + 2√(7-x) = 13, we can follow these steps:

Step 1: Identify the radical terms in the equation. In this case, we have two radical terms: √(5x-6) and √(7-x).

Step 2: Isolate one of the radical terms on one side of the equation. We'll choose to isolate √(5x-6). Subtracting 2√(7-x) from both sides, we get:
3√(5x-6) = 13 - 2√(7-x)

Step 3: Square both sides of the equation to eliminate the radical on the left-hand side. This yields:
(3√(5x-6))^2 = (13 - 2√(7-x))^2
9(5x-6) = (13 - 2√(7-x))(13 - 2√(7-x))

Step 4: Simplify the right-hand side of the equation by expanding the binomial. This results in:
45x - 54 = 169 - 26√(7-x) - 26√(7-x) + 4(7-x)
45x - 54 = 169 - 52√(7-x) + 28 - 4x

Step 5: Combine like terms on both sides of the equation:
45x + 4x - 54 + 4x + 52√(7-x) = 169 + 28

Step 6: Simplify:
53x - 54 + 52√(7-x) = 197

Step 7: Isolate the radical term by subtracting 52√(7-x) from both sides:
53x - 54 = 197 - 52√(7-x)

Step 8: Isolate the radical by squaring both sides of the equation:
(53x - 54)^2 = (197 - 52√(7-x))^2
(53x - 54)(53x - 54) = (197 - 52√(7-x))(197 - 52√(7-x))

Step 9: Simplify the right-hand side of the equation by expanding the binomial:
2809x^2 - 2844x + 2916 = 38809 - 4076√(7-x) + 2704(7-x)

Step 10: Combine like terms on both sides of the equation:
2809x^2 - 2844x + 2916 - 2704(7-x) = 38809 - 4076√(7-x)

Step 11: Simplify:
2809x^2 - 2844x + 2916 - 18928 + 2704x = 38809 - 4076√(7-x)

Step 12: Combine like terms:
2809x^2 - 1340x - 16012 = 38809 - 4076√(7-x)

Step 13: Move all terms to one side of the equation:
2809x^2 - 1340x - 38809 + 16012 + 4076√(7-x) = 0

Step 14: Simplify:
2809x^2 - 1340x - 22797 + 4076√(7-x) = 0

This quadratic equation does not have a simple method to solve for x. You can either try factoring, completing the square, or using the quadratic formula to find the values of x that satisfy this equation.