Solve the equation [tex]$\sqrt{x+4}-3=1$[/tex] for the variable. Show each step of your solution process.
Answer (2)
To solve the equation x + 4 − 3 = 1 , we first isolate the square root and then square both sides to find x = 12 . Finally, we check the solution and confirm it is correct. Thus, the answer is x = 12 .
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Isolate the square root term: x + 4 = 4 .
Square both sides: x + 4 = 16 .
Isolate x : x = 12 .
Check the solution: 12 + 4 − 3 = 1 , which simplifies to 1 = 1 . Thus, the solution is 12 .
Explanation
Understanding the Problem We are given the equation x + 4 − 3 = 1 and we want to solve for the variable x . Our goal is to isolate x by performing algebraic operations on both sides of the equation.
Isolating the Square Root First, we isolate the square root term by adding 3 to both sides of the equation: x + 4 − 3 + 3 = 1 + 3
x + 4 = 4
Squaring Both Sides Next, we square both sides of the equation to eliminate the square root: ( x + 4 ) 2 = 4 2
x + 4 = 16
Isolating x Now, we isolate x by subtracting 4 from both sides of the equation: x + 4 − 4 = 16 − 4
x = 12
Checking the Solution Finally, we check our solution by substituting x = 12 back into the original equation: 12 + 4 − 3 = 1
16 − 3 = 1
4 − 3 = 1
1 = 1
Since the equation holds true, our solution is valid.
Final Answer Therefore, the solution to the equation x + 4 − 3 = 1 is x = 12 .
Examples
Imagine you are building a square garden and need to determine the length of each side. You know that if you add 4 feet to the length of one side and then take the square root of that new length, subtracting 3 feet from the result will give you a final length of 1 foot. This problem helps you calculate the original length of the side of your garden. Solving equations with square roots is useful in various real-world scenarios, such as calculating distances, designing structures, and modeling physical phenomena.
