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Chapter 15: Problem 16

Solve and check. $$\sqrt{3 x}=4$$

Short Answer

Expert verified
The solution to the equation is \(x = 5.\overline{3}\)

Step by step solution

01

Remove the square root

Start by getting rid of the square root on the left side of the equation. This can be done by squaring both sides of the equation: \((\sqrt{3x})^2 = 4^2\) which simplifies to \(3x = 16\)
02

Isolate the variable

Now, isolate x by dividing both sides of the equation by 3: \(x = 16 / 3\) which simplifies to \(x = 5.\overline{3}\)
03

Check the solution

It is always a good practice to check the solution. Replace x with \(5.\overline{3}\) in the original equation, \(\sqrt{3(5.\overline{3})}=4\), which simplifies to \(4 = 4\). The left and right side are equal, therefore the obtained solution is accurate.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Radical Equations
Radical equations are equations in which the variable is under a radical symbol, such as a square root. Solving these equations involves isolating the radical and then eliminating it by performing operations.
In this exercise, we begin with the equation \(\sqrt{3x} = 4\), a classic example of a radical equation.
  • Firstly, identify the radical in your equation, like the square root in the example.
  • Your goal is to "undo" this square root by squaring both sides of the equation.
  • This process helps in removing the radical while maintaining the equality.
After the square root is eliminated, the problem becomes a simpler equation without radicals, making it easier to solve.
However, always remember to check your solutions to ensure they are valid in the context of the original equation.
Equation Simplification
Equation simplification is a key step in solving any algebraic equation, and in this case, it involves working with our newly structured equation after removing the radical.
  • In the example, squaring both sides transforms the equation into \(3x = 16\).
  • This step is crucial as it converts the radical equation into a simple linear equation.
  • Simplification allows us to isolate and solve for the variable, making the equation manageable.
The essence of simplification is reducing the equation to its most basic form, so you can clearly see the necessary steps to find the solution.
Remember, every action taken in simplification should maintain the original equation's balance or equality.
Algebraic Solutions
Once the equation is simplified, the next focus is on finding the algebraic solution, which involves isolating the variable of interest, in this case, \(x\).
  • Post-simplification, isolate the variable by performing operations such as addition, subtraction, multiplication, or division.
  • For instance, in our case, division is used to solve for \(x\), dividing both sides by 3.
  • This results in \(x = \frac{16}{3}\), or \(x = 5.\overline{3}\).
This approach leads to finding the value of the variable, hence solving the equation.
Finally, it is critical to check the solution by substituting it back into the original equation to verify that it satisfies the equation.
If both sides of the equation are equal after substitution, your solution is confirmed to be correct.
This verification ensures the correctness and reliability of your solution in solving any algebraic equation.

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