What is the solution of $\sqrt{x-4}+5=2 $?
Answer (2)
The solution to the equation x − 4 + 5 = 2 leads to an impossibility where the square root equals a negative number, indicating there is no real solution. After isolating the square root and squaring both sides, we find x = 13 , but this does not satisfy the original equation. Thus, the final conclusion is that there is no solution.
;
Isolate the square root: x − 4 = − 3 .
Square both sides: x − 4 = 9 .
Solve for x : x = 13 .
Check the solution: 13 − 4 + 5 = 8 = 2 . Therefore, there is no solution .
Explanation
Understanding the problem We are given the equation x − 4 + 5 = 2 . We need to find the value of x that satisfies this equation.
Isolating the square root First, we isolate the square root term by subtracting 5 from both sides of the equation: x − 4 = 2 − 5 x − 4 = − 3
Squaring both sides Now, we square both sides of the equation to eliminate the square root: ( x − 4 ) 2 = ( − 3 ) 2 x − 4 = 9
Solving for x Solve for x by adding 4 to both sides: x = 9 + 4 x = 13
Checking the solution We need to check if the solution is valid by substituting x = 13 into the original equation: 13 − 4 + 5 = 9 + 5 = 3 + 5 = 8 Since 8 = 2 , the solution x = 13 is not valid.
Final Answer Since the square root term x − 4 must be non-negative, and we have x − 4 = − 3 , which is impossible for any real number x , there is no real solution.
Examples
When solving problems involving distances or physical quantities that cannot be negative, we often encounter square roots. For example, calculating the distance between two points involves square roots, and distances cannot be negative. Similarly, in physics, the speed of an object is the square root of its kinetic energy divided by its mass ( m 2 K ). Since speed cannot be negative, we only consider the positive square root. Understanding how to solve equations with square roots and check for extraneous solutions is crucial in these real-world applications.
