To solve this equation, we first need to isolate the square roots on one side of the equation and square both sides to get rid of the square roots.
√(x+3) - √(7-x) = √(2x-8)
First, let's move one of the square roots to the other side of the equation:
√(x+3) = √(7-x) + √(2x-8)
Now, square both sides of the equation to eliminate the square roots:
(x+3) = (7-x) + 2√[(7-x)(2x-8)] + (2x-8)
Expanding the terms gives:
x + 3 = 7 - x + 2√[14x - 2x^2 - 16x + 64] + 2x - 8
Combining like terms and simplifying further:
x + 3 = 7 - x + 2√[14x - 2x^2 - 16x + 64] + 2x - x + 3 = -x + 6 + 2√[-2x^2 - 2x + 642x = 3 + 2√[64 - 2(x^2 + x)]
Now square both sides again to eliminate the square root:
(2x)^2 = (3 + 2√[64 - 2(x^2 + x)])^4x^2 = 9 + 12√[64 - 2(x^2 + x)] + 4(64 - 2(x^2 + x)4x^2 = 9 + 12√[64 - 2(x^2 + x)] + 256 - 8(x^2 + x)
Rearranging and simplifying gives:
4x^2 = 265 - 8x^2 - 4x + 12√[64 - 2(x^2 + x)12x^2 + 4x - 265 = 12√[64 - 2(x^2 + x)]
To continue solving for x, we would need to isolate the square root term and then square both sides once more. This process may result in a complex solution, so it's recommended to check for any extraneous solutions.
To solve this equation, we first need to isolate the square roots on one side of the equation and square both sides to get rid of the square roots.
√(x+3) - √(7-x) = √(2x-8)
First, let's move one of the square roots to the other side of the equation:
√(x+3) = √(7-x) + √(2x-8)
Now, square both sides of the equation to eliminate the square roots:
(x+3) = (7-x) + 2√[(7-x)(2x-8)] + (2x-8)
Expanding the terms gives:
x + 3 = 7 - x + 2√[14x - 2x^2 - 16x + 64] + 2x - 8
Combining like terms and simplifying further:
x + 3 = 7 - x + 2√[14x - 2x^2 - 16x + 64] + 2x -
x + 3 = -x + 6 + 2√[-2x^2 - 2x + 64
2x = 3 + 2√[64 - 2(x^2 + x)]
Now square both sides again to eliminate the square root:
(2x)^2 = (3 + 2√[64 - 2(x^2 + x)])^
4x^2 = 9 + 12√[64 - 2(x^2 + x)] + 4(64 - 2(x^2 + x)
4x^2 = 9 + 12√[64 - 2(x^2 + x)] + 256 - 8(x^2 + x)
Rearranging and simplifying gives:
4x^2 = 265 - 8x^2 - 4x + 12√[64 - 2(x^2 + x)
12x^2 + 4x - 265 = 12√[64 - 2(x^2 + x)]
To continue solving for x, we would need to isolate the square root term and then square both sides once more. This process may result in a complex solution, so it's recommended to check for any extraneous solutions.