$\sqrt{3 x+4}=1+\sqrt{3 x-11}$
When both sides of the equation are squared, the resulting equation is
A. $3 x+4=9 x^2+122$
B. $3 x+4=3 x-10$
C. $3 x+4=2 \sqrt{3 x-11}+3 x-10$
Answer (2)
After squaring both sides of the equation 3 x + 4 = 1 + 3 x − 11 , we derived the equation 3 x + 4 = 2 3 x − 11 + 3 x − 10 . This matches with option C among the given choices. The final answer is option C.
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Square both sides of the original equation: ( 3 x + 4 ) 2 = ( 1 + 3 x − 11 ) 2 .
Expand the right side: 3 x + 4 = 1 + 2 3 x − 11 + ( 3 x − 11 ) .
Simplify the equation: 3 x + 4 = 3 x − 10 + 2 3 x − 11 .
The correct equation is: 3 x + 4 = 2 3 x − 11 + 3 x − 10 .
Explanation
Initial Equation We are given the equation 3 x + 4 = 1 + 3 x − 11 . Our goal is to determine the correct equation after squaring both sides.
Squaring Both Sides Squaring both sides of the original equation, we get: ( 3 x + 4 ) 2 = ( 1 + 3 x − 11 ) 2
Expanding the Right Side Expanding the right side of the equation, we have: 3 x + 4 = 1 + 2 3 x − 11 + ( 3 x − 11 ) 3 x + 4 = 1 + 2 3 x − 11 + 3 x − 11 3 x + 4 = 3 x − 10 + 2 3 x − 11
Comparing with Given Options Comparing the derived equation with the given options:
The options are:
3 x + 4 = 9 x 2 + 122
3 x + 4 = 3 x − 10
3 x + 4 = 2 3 x − 11 + 3 x − 10
Our derived equation is 3 x + 4 = 3 x − 10 + 2 3 x − 11 , which is the same as option 3.
Final Answer Therefore, the correct equation after squaring both sides of the original equation is: 3 x + 4 = 2 3 x − 11 + 3 x − 10
Examples
When solving equations involving square roots, it's crucial to square both sides to eliminate the radicals. This technique is widely used in physics to solve problems related to energy and momentum, where quantities are often expressed as square roots. For instance, determining the velocity of an object based on its kinetic energy involves squaring both sides of an equation to isolate the velocity term. This algebraic manipulation simplifies the equation and allows for a straightforward solution.
