This equation contains square roots, and solving it requires eliminating the radicals by squaring both sides step by step. The equation is:

$\sqrt{x^2 - 5x + 4} + \sqrt{2x - 3} = 7$

Steps to solve:

1. Isolate one square root term: Subtract $\sqrt{2x - 3}$ from both sides: $\sqrt{x^2 - 5x + 4} = 7 - \sqrt{2x - 3}$

2. Square both sides to eliminate the first radical: Squaring both sides: $x^2 - 5x + 4 = (7 - \sqrt{2x - 3})^2$ Expanding the right-hand side: $x^2 - 5x + 4 = 49 - 14\sqrt{2x - 3} + (2x - 3)$ Simplify: $x^2 - 5x + 4 = 46 + 2x - 14\sqrt{2x - 3}$

3. Isolate the remaining square root term: Move all terms except the square root to one side: $14\sqrt{2x - 3} = 46 + 2x - (x^2 - 5x + 4)$ Simplify: $14\sqrt{2x - 3} = 46 + 2x - x^2 + 5x - 4$ $14\sqrt{2x - 3} = -x^2 + 7x + 42$

4. Square both sides again: To remove the remaining square root: $(14)^2 (2x - 3) = (-x^2 + 7x + 42)^2$ Expand both sides and solve for $x$. This results in a polynomial equation, which can then be solved using standard techniques (factoring, quadratic formula, etc.).

Would you like me to expand this further and simplify the polynomial explicitly? Let me know!

Follow-up Questions:

1. Would you like me to show every step of the expansion?
2. Should we confirm potential solutions by substituting them back into the original equation?
3. Do you need clarification on why we square both sides twice?
4. Would you like a numerical approximation for the solutions?
5. Should I explain any specific algebraic rule applied here?

Tip:

When solving equations involving square roots, always verify your solutions to ensure no extraneous roots arise from squaring.