Solve the radical equation $x-1=\sqrt{x-1}$. Check the correctness of your solutions.
Answer (1)
Substitute y = x − 1 to simplify the equation to y = y .
Square both sides to get y 2 = y , then rearrange to y 2 − y = 0 .
Factor to find solutions for y : y = 0 or y = 1 .
Substitute back to find solutions for x : x = 1 and x = 2 , both of which are valid. Thus, the solutions are 1 , 2 .
Explanation
Understanding the Problem We are given the radical equation x − 1 = x − 1 . Our goal is to find all possible values of x that satisfy this equation and then verify that these solutions are indeed correct.
Simplifying with Substitution To make the equation easier to work with, let's make a substitution. Let y = x − 1 . Then our equation becomes y = y . This is a simpler equation to solve.
Eliminating the Square Root Now, let's square both sides of the equation y = y to eliminate the square root. Squaring both sides gives us y 2 = ( y ) 2 , which simplifies to y 2 = y .
Solving the Quadratic Equation Next, we want to solve the quadratic equation y 2 = y . To do this, we first rearrange the equation to get y 2 − y = 0 . Then, we factor the left side to obtain y ( y − 1 ) = 0 .
Finding the Values of y From the factored equation y ( y − 1 ) = 0 , we see that the solutions for y are y = 0 or y − 1 = 0 , which means y = 0 or y = 1 .
Finding the Values of x Now we need to substitute back x − 1 for y to find the solutions for x . If y = 0 , then x − 1 = 0 , so x = 1 . If y = 1 , then x − 1 = 1 , so x = 2 . Thus, our possible solutions for x are x = 1 and x = 2 .
Checking the Solutions Finally, we need to check if these solutions are correct by plugging them back into the original equation x − 1 = x − 1 .
For x = 1 , we have 1 − 1 = 1 − 1 , which simplifies to 0 = 0 , or 0 = 0 . This is true, so x = 1 is a solution.
For x = 2 , we have 2 − 1 = 2 − 1 , which simplifies to 1 = 1 , or 1 = 1 . This is also true, so x = 2 is a solution.
Therefore, the solutions to the equation are x = 1 and x = 2 .
Examples
Radical equations appear in various contexts, such as physics and engineering. For instance, when calculating the speed of an object falling under gravity through a medium with resistance, the equation might involve square roots. Solving these equations helps determine critical parameters like terminal velocity or the time it takes to reach a certain speed. Understanding how to manipulate and solve radical equations is essential for making accurate predictions and designing systems effectively.
