To solve the equation (5x-3)^2 - (4x-7)^2 = 49, we can expand the squared terms and simplify the equation:

(5x - 3)^2 - (4x - 7)^2 = 49
(25x^2 - 30x + 9) - (16x^2 - 56x + 49) = 49
25x^2 - 30x + 9 - 16x^2 + 56x - 49 = 49
9x^2 + 26x - 40 = 49
9x^2 + 26x - 89 = 0

Now we have a quadratic equation. We can solve it by factoring, completing the square, or using the quadratic formula. In this case, the quadratic equation does not factor easily, so we'll use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

For the equation 9x^2 + 26x - 89 = 0, the coefficients are:
a = 9, b = 26, c = -89

Substituting these values into the quadratic formula:

x = (-26 ± √(26^2 - 4 * 9 * -89)) / (2 * 9)
x = (-26 ± √(676 + 3204)) / 18
x = (-26 ± √(3880)) / 18
x = (-26 ± √(4 * 970)) / 18
x = (-26 ± 2√(970)) / 18
x = (-13 ± √(970)) / 9

So the solutions to the equation are:
x = (-13 + √(970)) / 9
x = (-13 - √(970)) / 9