√7-2x-√3-x²=√-x

To solve the equation √(7 - 2x) - √(3 - x²) = √(-x), we start by isolating one of the square root terms. Let us first transfer the √(3 - x²) to the other side of the equation.

√(7 - 2x) = √(3 - x²) + √(-x)

Now square both sides of the equation to eliminate one of the square roots:

(√(7 - 2x))^2 = (√(3 - x²) + √(-x))^2

When you square the left-hand side, the square root is eliminated, and we are left with the expression inside the square root:

7 - 2x = (√(3 - x²) + √(-x))^2

Expanding the right side using the formula (a + b)^2 = a^2 + 2ab + b^2, we get:

7 - 2x = (√(3 - x²))^2 + 2√((3 - x²)(-x)) + (√(-x))^2

Now, simplify the right side:

7 - 2x = 3 - x² + 2√(3 - x²)(-x) + (-x)

Bring the terms not involving the square root to the left-hand side of the equation:

7 - 2x - 3 + x² + x = 2√(3 - x²)(-x)

Combine like terms:

x² - x + 4 = 2√(3 - x²)(-x)

To eliminate the square root, we square both sides again:

(x² - x + 4)^2 = (2√(3 - x²)(-x))^2

Expand the left side using binomial expansion:

(x² - x + 4)(x² - x + 4) = 4(3 - x²)(-x)

This will give us a quartic equation after simplifying, and the process could become more complex. So, let's review the steps taken as there might be an algebraic simplification missed that would preclude having to deal with a quartic equation.

Upon closer inspection at the initial equation and the steps followed, it appears that there might have been a more straightforward approach or a mistake. For now, since we do not have a quartic solver, we might look for alternative methodologies such as graphical solutions, approximation techniques, or confirming that the initial equation format was correctly manipulated.

It might help to graph or use numerical methods to find the solution for a complicated equation like this. However, if you need to solve it analytically, the process may be quite involved and difficult to manage without more sophisticated mathematical tools or software.