%5E1%2F2-(-2-2x)%5E1%2F2%3D1.jpeg "(-3-4x)^1/2-(-2-2x)^1/2=1")
Solving the Equation: √(-3-4x) - √(-2-2x) = 1
This equation involves square roots, making it a bit tricky to solve directly. Here's how we can approach it:
1. Isolate one of the square roots:
Start by moving one of the square root terms to the right side of the equation:
√(-3-4x) = 1 + √(-2-2x)
2. Square both sides:
Squaring both sides will eliminate one of the square roots:
(-3-4x) = (1 + √(-2-2x))^2
3. Expand and simplify:
Expand the right side of the equation:
-3 - 4x = 1 + 2√(-2-2x) + (-2-2x)
Simplify:
-4x - 4 = 2√(-2-2x) - 2x - 1
Further simplification:
-2x - 3 = 2√(-2-2x)
4. Square both sides again:
Square both sides of the equation to eliminate the remaining square root:
4x^2 + 12x + 9 = 4(-2 - 2x)
5. Solve the quadratic equation:
Expand and rearrange to get a standard quadratic equation:
4x^2 + 20x + 17 = 0
This quadratic equation can be solved using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
Where a = 4, b = 20, and c = 17.
6. Check for extraneous solutions:
After finding the solutions for x, remember to check them in the original equation to ensure they are valid. Sometimes, squaring both sides can introduce extraneous solutions, which are solutions that don't satisfy the original equation.
Note:
It's important to note that the square root of a negative number is not defined in the real number system. Therefore, the expressions inside the square roots, (-3-4x) and (-2-2x), must be non-negative for the solutions to be valid. This will restrict the domain of the solutions.