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Chapter 15: Problem 17

Solve and check. $$\sqrt{4 x}=8$$

Short Answer

Expert verified
The solution to the equation \( \sqrt{4 x} = 8 \) is \(x = 16\).

Step by step solution

01

Eliminate the square root

To eliminate the square root on the left side of the equation, squaring both sides of the equation is the best choice. This means \((\sqrt{4x})^2 = 8^2\) which simplifies to \(4x = 64\).
02

Solving the resulted linear equation

We're now dealing with a simple linear equation in the form of \(4x=64\). We can solve for x by dividing both sides of the equation by 4. That is, \(x = \frac{64}{4}\). This gives \(x = 16\).
03

Check the solution

Now, we will check if \(x = 16\) is the right solution by plugging it back into the original equation. We substitute \(x = 16\) into the original equation \(\sqrt{4 x} = 8\), we get \(\sqrt{4*16} = 8\), which simplifies to \(8=8\). Since the left side is equal to the right side, this confirms that the solution is indeed \(x = 16\).

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Key Concepts

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

Linear Equations
A linear equation is an equation where the highest power of the variable is one. In the step-by-step solution, after eliminating the square root, we encounter a linear equation in the form of \(4x = 64\). This is a simple linear equation where we only need one operation to isolate the variable \(x\).

To solve this linear equation, we divide both sides by the coefficient of \(x\), which is 4. This operation "undoes" the multiplication by 4, leaving \(x\) by itself on one side.

When solving linear equations, always remember:
  • To perform the same operation on both sides of the equation.
  • To keep balance, similar to a scale.
  • To work step-by-step until the variable is isolated.
This process will lead you neatly to the solution of the equation.
Checking Solutions
Once you've found a solution, it's crucial to check if it's correct. This step ensures that the solution you've obtained truly satisfies the original equation. In our problem, after finding \(x = 16\), we plug it back into the original equation \(\sqrt{4x} = 8\).

Checking involves substituting the value of \(x\) back and simplifying both sides of the equation to see if they are equal. Let's break it down:
  • Substitute \(x = 16\) into the equation, yielding \(\sqrt{4 \times 16} = 8\).
  • Simplify inside the square root to get \(\sqrt{64} = 8\).
  • Since \(\sqrt{64} = 8\) is true, both sides are equal.
This confirms that \(x = 16\) is a valid solution. Always remember, checking solutions help catch any possible errors made during calculations.
Simplifying Equations
Simplifying equations is about making an equation easier to understand and solve. In the given exercise, after eliminating the square root, you have the equation \(4x = 64\). The goal is to simplify this equation to more clearly see what \(x\) equals.

To simplify, you'll generally:
  • Combine like terms.
  • Perform operations to isolate the variable.
  • Use inverse operations to "undo" operations around the variable.
In our case, simplifying involves dividing both sides by 4, leading to \(x = 16\). Simplification transforms complex-looking expressions into simpler ones, making it easier to identify solutions.

Remember, the simpler the equation, the fewer steps needed to solve it. This task requires careful arithmetic, ensuring that each step maintains equality in the equation.

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