Answer:
x = 13.
Explanation:
To solve the equation √(2x-1) - √(x+3) = 1, we can follow these steps:
Start by isolating one of the square roots on one side of the equation. We can do this by adding √(x+3) to both sides:
√(2x-1) = 1 + √(x+3)
Now we can square both sides of the equation to eliminate the square roots:
(√(2x-1))^2 = (1 + √(x+3))^2
Simplifying both sides, we get:
2x - 1 = 1 + 2√(x+3) + x + 3
Next, we can simplify the right side of the equation by combining like terms:
2x - 1 = x + 4 + 2√(x+3)
We can then isolate the square root on one side of the equation by subtracting x+4 from both sides:
x - 5 = 2√(x+3)
To isolate x, we can square both sides of the equation again:
(x - 5)^2 = (2√(x+3))^2
Simplifying both sides, we get:
x^2 - 10x + 25 = 4x + 12
Moving all the terms to one side of the equation and simplifying, we get:
x^2 - 14x + 13 = 0
Factoring the quadratic equation, we get:
(x - 1)(x - 13) = 0
Therefore, the solutions to the equation are x = 1 and x = 13.
However, we need to check our solutions to ensure that they do not result in any extraneous solutions, which can arise from the process of squaring both sides of the equation. We can plug each solution back into the original equation and verify that it is a valid solution:
When x = 1:
√(2(1)-1) - √(1+3) = 1
√1 - √4 = 1
1 - 2 = 1 (This is false, so x = 1 is not a solution)
When x = 13:
√(2(13)-1) - √(13+3) = 1
√25 - √16 = 1
5 - 4 = 1 (This is true, so x = 13 is a valid solution)
Therefore, the only solution to the equation is x = 13.