Problem 29 - Algebra A Combined Function

Question

Solve. $$ \sqrt[4]{4 x+1}-2=0 $$

Step-by-Step Solution

Answer

The solution is \( x = \frac{15}{4} \).

Step 1: Isolate the Radical

First, let's isolate the radical expression on one side of the equation. We start with the original equation:\[ \sqrt[4]{4x+1} - 2 = 0 \]To isolate the radical, add 2 to both sides:\[ \sqrt[4]{4x+1} = 2 \]

Step 2: Eliminate the Radical

To eliminate the fourth root, raise both sides of the equation to the power of 4:\[ (\sqrt[4]{4x+1})^4 = 2^4 \]This simplifies to:\[ 4x + 1 = 16 \]

Step 3: Solve for x

Now, solve the equation for \( x \). Subtract 1 from both sides:\[ 4x = 15 \]Then, divide both sides by 4 to solve for \( x \):\[ x = \frac{15}{4} \]

Key Concepts

Understanding Radical Equations

Radical equations are equations in which the variable is located inside a radical sign. In simple terms, when you see a square root, cube root, or higher roots in the equation, you're dealing with a radical equation. These types of equations can initially look quite complicated, but their solutions follow a methodical approach.

Radical equations are often identified by their root signs, which could include:

• Square roots, denoted by \( \sqrt{ } \)
• Cube roots as \( \sqrt[3]{ } \)
• Fourth roots or higher such as \( \sqrt[4]{ } \)

Solving radical equations requires careful steps, primarily to ensure that we correctly find the value of the variable hidden within the radical sign. Not following the proper steps could lead to errors or incomplete solutions, which is why understanding each procedure is essential.

Isolation of Radicals

The first crucial step in solving radical equations is isolating the radical expression. This means rearranging the equation so that the term with the radical stands alone on one side of the equation. By doing this, you simplify the process because it makes the next steps, such as eliminating the radical, more straightforward.

For example, consider the equation \( \sqrt[4]{4x+1} - 2 = 0 \). To isolate the radical, you add 2 to both sides, resulting in \( \sqrt[4]{4x+1} = 2 \). Now, the radical is isolated and easier to manage in the subsequent steps.

This process of isolation is crucial, as it sets the stage for accurately solving the equation without getting incorrect results.

Solving Equations with Isolated Radicals

Once you have isolated the radical in a radical equation, the next step is to eliminate it. This often involves raising both sides of the equation to a power that corresponds with the type of root.

• If dealing with a square root, you'll square both sides.
• For a cube root, cube both sides, and so forth.

In our equation, \( \sqrt[4]{4x+1} = 2 \), the fourth root requires raising both sides to the power of 4:
\( (\sqrt[4]{4x+1})^4 = 2^4 \).

This eliminates the radical, simplifying the equation to \( 4x + 1 = 16 \). Now, solving the simplified equation becomes straightforward:

• Subtract 1 from both sides to get \( 4x = 15 \).
• Finally, divide both sides by 4 to find \( x = \frac{15}{4} \).

Following these steps will lead you to the correct solution consistently, ensuring that all parts of the equation are correctly handled.