Question

Solve the equation. Check for extraneous solutions. $$\sqrt{4 x}-1=3$$

Short answer

The solution to the equation is \( x = 4 \).

Step-by-step solution

Isolate the square root

For the equation \( \sqrt{4x} - 1 = 3 \), first add 1 to both sides of the equation. This gives: \( \sqrt{4x} = 4 \).

Remove the square root

Square both sides of the equation to eliminate the square root. Thus, the equation becomes: \( {(\sqrt{4x})}^2 = 4^2 \) or \( 4x = 16 \).

Solve for x

Now, divide both sides of the equation by 4 to isolate x, yielding \( x = 4 \).

Check for extraneous solutions

Substitute 4 in place of x into the original equation to verify the solution: \( \sqrt{4(4)} - 1 = 3 \), which simplifies to \( 4 - 1 = 3 \). Since both sides of the equation are equal, we can conclude that x=4 is a valid solution to the equation.

Key concepts

Isolate the Square Root

When solving equations that involve square roots, the first step is to isolate the square root on one side of the equation. This means we need to move all the other terms to the opposite side. For instance, in the equation
\( \sqrt{4x} - 1 = 3 \),
the goal is to have the square root term, \( \sqrt{4x} \), by itself. To achieve this, you would perform basic algebraic operations such as adding or subtracting terms on both sides of the equation. Here, we add 1 to each side, resulting in
\( \sqrt{4x} = 4 \).
Isolating the square root sets the stage for the next step, which is to remove the square root. Ensuring that the square root stands alone makes it easier to deal with and paves the way for finding the actual solution for \( x \).

Remove the Square Root

Having isolated the square root, we then proceed to 'remove' it. This is done by performing an operation that negates the square root, known as squaring. Squaring the square root (and the entire other side of the equation to keep the equation balanced) will result in the root being removed, because the square root and the square are opposite operations. For the equation
\( \sqrt{4x} = 4 \),
we square both sides, leading us to
\( {(\sqrt{4x})}^2 = 4^2 \),
which simplifies to \( 4x = 16 \).
It is crucial to square the entire sides of the equation, not just the terms, to adhere to algebraic principles and to reach the correct simplified form of the equation.

Check for Extraneous Solutions

Even after finding values that seem to be solutions, it is important to check for extraneous solutions. These are 'solutions' that satisfy the algebraic form of the equation but do not actually solve the original equation. To check for extraneous solutions, substitute the found values back into the original equation. In our example, we substitute \( x = 4 \) back into the original equation \( \sqrt{4x} - 1 = 3 \):
\( \sqrt{4(4)} - 1 = 3 \),
which simplifies to \( 4 - 1 = 3 \).
Since this check results in a true statement, the solution \( x = 4 \) is valid. If the statement was false, the solution would be extraneous and therefore incorrect. Always verify your solutions to prevent including incorrect answers in your final results.