Select whether the equation has a solution or not.
[tex]$\sqrt{x-5}=4$[/tex]
A. no roots
B. roots
Answer (2)
Square both sides of the equation: ( x − 5 ) 2 = 4 2 , which simplifies to x − 5 = 16 .
Solve for x : x = 16 + 5 = 21 .
Check if the solution is valid: 21 − 5 = 16 ≥ 0 , which is true.
The equation has a solution: roots .
Explanation
Understanding the Problem We are given the equation x − 5 = 4 . Our goal is to determine whether this equation has a solution or not. To do this, we will solve for x and check if the solution is valid.
Squaring Both Sides To solve for x , we square both sides of the equation: ( x − 5 ) 2 = 4 2 This simplifies to: x − 5 = 16
Solving for x Now, we isolate x by adding 5 to both sides of the equation: x = 16 + 5 x = 21
Checking the Solution We need to check if our solution is valid. The original equation contains a square root, so we must ensure that the expression inside the square root is non-negative. In this case, we need to check if x − 5 ≥ 0 . Substituting x = 21 , we get: 21 − 5 = 16 ≥ 0 Since 16 is greater than or equal to 0 , our solution is valid.
Conclusion Since we found a valid solution for x , the equation x − 5 = 4 has a solution.
Examples
Imagine you are building a square garden and want to know the length of each side. If the area of the garden is represented by the equation x − 5 = 4 , where x is related to the area, solving this equation helps you find the exact value of x needed to determine the side length. This type of problem is useful in various scenarios, such as calculating dimensions in construction, determining speeds in physics, or even in financial calculations.
The equation x − 5 = 4 has a solution, which is x = 21 . This solution is valid as the expression inside the square root is non-negative. Therefore, the correct answer is B. roots.
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