Select whether the equation has a solution or not.
[tex]$\sqrt{12 x-23}=\sqrt{3 x-11}+\sqrt{3 x}$[/tex]
A. roots
B. no roots
Answer (2)
Square both sides of the equation: ( 12 x − 23 ) 2 = ( 3 x − 11 + 3 x ) 2 , which simplifies to 12 x − 23 = 6 x − 11 + 2 9 x 2 − 33 x .
Isolate the square root term and divide by 2: 3 x − 6 = 9 x 2 − 33 x .
Square both sides again: ( 3 x − 6 ) 2 = 9 x 2 − 33 x , which simplifies to 9 x 2 − 36 x + 36 = 9 x 2 − 33 x .
Solve for x: − 3 x = − 36 , so x = 12 . Check that x = 12 is indeed a solution. Thus, the equation has a solution.
Explanation
Problem Analysis We are given the equation 12 x − 23 = 3 x − 11 + 3 x and we need to determine if it has a solution.
Squaring Both Sides First, let's square both sides of the equation to eliminate the square roots: ( 12 x − 23 ) 2 = ( 3 x − 11 + 3 x ) 2
12 x − 23 = ( 3 x − 11 ) + 2 ( 3 x − 11 ) ( 3 x ) + 3 x 12 x − 23 = 6 x − 11 + 2 9 x 2 − 33 x Now, isolate the square root term: 6 x − 12 = 2 9 x 2 − 33 x Divide both sides by 2: 3 x − 6 = 9 x 2 − 33 x
Squaring Again Square both sides again to eliminate the remaining square root: ( 3 x − 6 ) 2 = ( 9 x 2 − 33 x ) 2 9 x 2 − 36 x + 36 = 9 x 2 − 33 x Simplify the equation: − 3 x = − 36 x = 12
Checking the Solution Now, we need to check if the solution x = 12 satisfies the original equation and the non-negativity conditions for the terms inside the square roots:
12 x − 23 ≥ 0 ⇒ 12 ( 12 ) − 23 = 144 − 23 = 121 ≥ 0 (True)
3 x − 11 ≥ 0 ⇒ 3 ( 12 ) − 11 = 36 − 11 = 25 ≥ 0 (True)
3 x ≥ 0 ⇒ 3 ( 12 ) = 36 ≥ 0 (True) Now, substitute x = 12 into the original equation: 12 ( 12 ) − 23 = 3 ( 12 ) − 11 + 3 ( 12 ) 144 − 23 = 36 − 11 + 36 121 = 25 + 36 11 = 5 + 6 11 = 11 The solution x = 12 satisfies the original equation and all non-negativity conditions.
Conclusion Since the solution x = 12 satisfies the original equation and the non-negativity conditions, the equation has a solution.
Examples
When designing structures or mechanical systems, engineers often encounter equations involving square roots. For example, calculating the tension in cables or the stress on materials may lead to equations similar to the one we solved. Verifying whether such equations have real solutions is crucial to ensure the physical feasibility and stability of the design. In finance, similar equations can arise when modeling investment returns or calculating risk factors, where the existence of real solutions indicates the viability of a particular investment strategy. Therefore, understanding how to solve and verify the solutions of equations with square roots is a valuable skill in various fields.
The equation 12 x − 23 = 3 x − 11 + 3 x has a solution, specifically x = 12 , which satisfies the original equation and all non-negativity conditions. Therefore, the answer is 'A. roots'.
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