Solve for all possible values of x.
[tex]\sqrt{x+6}+5=17[/tex]
Answer (2)
To solve the equation x + 6 + 5 = 17 , we isolate the square root and square both sides to find x = 138 . After confirming the solution by substituting it back into the original equation, we establish that the solution is valid. The final answer is x = 138 .
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Isolate the square root term: x + 6 = 17 − 5 = 12 .
Square both sides: x + 6 = 1 2 2 = 144 .
Solve for x : x = 144 − 6 = 138 .
The solution is 138 .
Explanation
Understanding the Problem We are given the equation x + 6 + 5 = 17 and asked to solve for x .
Isolating the Square Root First, we isolate the square root by subtracting 5 from both sides of the equation: x + 6 + 5 − 5 = 17 − 5 x + 6 = 12
Squaring Both Sides Next, we square both sides of the equation to eliminate the square root: ( x + 6 ) 2 = 1 2 2 x + 6 = 144
Solving for x Now, we solve for x by subtracting 6 from both sides of the equation: x + 6 − 6 = 144 − 6 x = 138
Checking the Solution We should check our solution by substituting x = 138 back into the original equation: 138 + 6 + 5 = 144 + 5 = 12 + 5 = 17
Since this is true, our solution is valid.
Final Answer Therefore, the solution is x = 138 .
Examples
Imagine you are designing a square garden and need to determine the length of each side. If the area of the garden plus an additional 6 square feet must equal 144 square feet, you can use the equation x + 6 = 12 to find the side length, x , of the garden. Solving such equations helps in practical scenarios like designing layouts, calculating dimensions, and managing space efficiently.
