Solve: [tex]$\sqrt{x-1}=\sqrt{x}=\frac{2}{\sqrt{x}}$[/tex]
Answer (2)
The equation x − 1 = x = x 2 is unsolvable because testing the only candidate solution x = 2 does not satisfy the initial conditions of the equation. Thus, there are no values of x that can solve the equation. Therefore, the conclusion is that there is no solution.
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Solve x = x 2 to find x = 2 .
Check if x = 2 satisfies x − 1 = x .
Since 1 = 2 , x = 2 is not a solution.
Therefore, there is no solution: No solution
Explanation
Understanding the Problem We are given the equation x − 1 = x = x 2 . We need to find the value(s) of x that satisfy this equation. Since we have square roots, we must have x ≥ 0 and x − 1 ≥ 0 , which means x ≥ 1 .
Solving for x From the equation x = x 2 , we can solve for x . Multiplying both sides by x gives us x = 2 .
Checking the Solution Now we need to check if x = 2 satisfies the original equation x − 1 = x . Substituting x = 2 into the equation x − 1 = x gives us 2 − 1 = 2 , which simplifies to 1 = 2 or 1 = 2 .
Conclusion Since 1 = 2 , x = 2 is not a solution. Therefore, there is no solution to the equation.
Examples
Consider a scenario where you are designing a suspension bridge. The equation x − 1 = x = x 2 might represent a simplified model of the forces and tensions acting on the bridge cables. Solving such equations helps engineers determine the stability and safety of the bridge structure. Although this specific equation might not directly apply, the underlying mathematical principles are crucial in structural engineering to ensure that bridges can withstand various loads and environmental conditions.
