Final answer:
The equation √(5-y)² = y-5 has one valid root, which is y = 10. This was found by first squaring both sides, then solving the resulting quadratic equation and verifying the potential roots in the original equation.
Step-by-step explanation:
To find the roots of the equation √(5-y)² = y-5, we start by removing the square root by squaring both sides of the equation, resulting in 5-y = (y-5)². Expanding the right side gives us 5-y = y² - 10y + 25. Rearranging terms to one side gives us y² - 11y + 20 = 0. This factors to (y-1)(y-10) = 0, leading to two possible solutions for y: y = 1 and y = 10. However, we need to verify these solutions in the original equation to ensure they are valid since squaring both sides can introduce extraneous solutions.
Upon substitution, we find that y = 1 does not satisfy the original equation while y = 10 does. Therefore, the only root of the original equation is y = 10.