Solve the radical equation. Check all proposed solutions.
$\sqrt{3 x}+11=x+5$
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The solution set is {$\square$}.
(Use a comma to separate answers as needed. Simplify your answer.)
B. The solution set is $\varnothing$.
Answer (1)
Isolate the radical: 3 x = x − 6 .
Square both sides: 3 x = x 2 − 12 x + 36 .
Solve the quadratic equation: x 2 − 15 x + 36 = 0 , which factors to ( x − 3 ) ( x − 12 ) = 0 , giving x = 3 or x = 12 .
Check for extraneous solutions: x = 3 is extraneous, and x = 12 is valid. The solution set is 12 .
Explanation
Problem Analysis We are given the radical equation 3 x + 11 = x + 5 . Our goal is to solve for x and check for any extraneous solutions that may arise from squaring both sides of the equation.
Isolating the Radical First, we isolate the radical term by subtracting 11 from both sides of the equation: 3 x = x + 5 − 11 3 x = x − 6
Squaring Both Sides Next, we square both sides of the equation to eliminate the square root: ( 3 x ) 2 = ( x − 6 ) 2 3 x = x 2 − 12 x + 36
Rearranging into Quadratic Form Now, we rearrange the equation into a quadratic equation by subtracting 3 x from both sides: 0 = x 2 − 12 x − 3 x + 36 0 = x 2 − 15 x + 36
Solving the Quadratic Equation We can solve this quadratic equation by factoring. We are looking for two numbers that multiply to 36 and add up to -15. These numbers are -3 and -12. So, we can factor the quadratic as follows: ( x − 3 ) ( x − 12 ) = 0 This gives us two possible solutions: x = 3 or x = 12 .
Checking for Extraneous Solutions Now, we need to check these proposed solutions in the original equation to see if they are valid or extraneous.
Checking x=3 Let's check x = 3 : 3 ( 3 ) + 11 = 3 + 5 9 + 11 = 8 3 + 11 = 8 14 = 8 This is false, so x = 3 is an extraneous solution.
Checking x=12 Now, let's check x = 12 : 3 ( 12 ) + 11 = 12 + 5 36 + 11 = 17 6 + 11 = 17 17 = 17 This is true, so x = 12 is a valid solution.
Final Answer Therefore, the solution set is {12}.
Examples
Radical equations are often used in physics to model various phenomena, such as the velocity of an object in free fall or the period of a pendulum. For example, if you are designing a suspension bridge, you might use radical equations to calculate the tension in the cables based on the bridge's length and the weight it needs to support. Understanding how to solve these equations ensures the bridge's stability and safety.
