Problem 9 - Elementary and Intermediate Algebra

Question

What is the first step in solving each equation? a. \(\sqrt{x+11}=5\) b. \(\sqrt[4]{5 x+4}+3-30\) c. \(\sqrt{x+8}-\sqrt{2 x}=1\)

Step-by-Step Solution

Answer

a. Isolate the square root term; b. Simplify the equation; c. Isolate one square root term.

Step 1a: Isolate the Square Root Term

For the equation \( \sqrt{x+11} = 5 \), the square root term is already isolated. This means we can proceed to the next step of removing the square root by squaring both sides of the equation.

Step 1b: Simplify the Equation

For the equation \( \sqrt[4]{5x+4} + 3 - 30 \), start by simplifying the equation. Combine like terms, which changes the equation to \( \sqrt[4]{5x+4} - 27 \), and then rewrite or isolate the root for solving.

Step 1c: Isolate One of the Square Root Terms

For the equation \( \sqrt{x+8} - \sqrt{2x} = 1 \), isolate one of the square root terms by adding \( \sqrt{2x} \) to both sides, giving \( \sqrt{x+8} = 1 + \sqrt{2x} \). Now the equation is set up to remove at least one square root in the next steps by squaring both sides.

Key Concepts

Square Root Isolation

Solving an equation that includes a square root begins with isolating the square root. This means getting the square root term by itself on one side of the equation. Doing this effectively prepares the equation for further operations.
For instance, if you have the equation \( \sqrt{x+11} = 5 \), the square root is already isolated as there are no other terms with it. Simplicity at this stage sets a straightforward path forward.
However, consider \( \sqrt{x+8} - \sqrt{2x} = 1 \), here you have two square root terms. You can isolate one term by adding \( \sqrt{2x} \) to both sides, resulting in \( \sqrt{x+8} = 1 + \sqrt{2x} \). This move leads to an equation where at least one root is by itself, making the next steps more manageable.

Equation Simplification

Before diving into deeper operations, simplifying the given equation is vital. Simplification sometimes involves combining like terms or rearranging elements to achieve a cleaner form.
Take the equation \( \sqrt[4]{5x+4} + 3 - 30 \) as an example. Start by simplifying within the equation to get \( \sqrt[4]{5x+4} - 27 \). Bringing like terms together reduces complexity, paving the way to easily isolate and manipulate the root term in subsequent steps.

• Combine similar terms: reducing arithmetic expressions.
• Rewrite the equation: getting it into a standard, simpler form helps in resolving further steps efficiently.

Squaring Both Sides

Once the square root term is isolated or the equation is simplified, the next step typically involves squaring both sides. This operation helps eliminate the square root, converting it into a more manageable algebraic expression.
For example, if faced with \( \sqrt{x+11} = 5 \), squaring both sides results in the expression \( x+11 = 25 \). Similarly, when dealing with an equation like \( \sqrt{x+8} = 1 + \sqrt{2x} \), you square each side separately. Be mindful that squaring can introduce extraneous solutions, so each answer must be checked in the original equation.

Radical Equations Methods

Solving radical equations effectively requires a blend of systematic methods and careful checks along the way. Applying these methods ensures that the solutions obtained are both valid and accurate.
A common strategy includes:

• Isolation and Simplification: as discussed, this gets to the core of the primary operations required.
• Checking Solutions: verify each solution derived from squaring both sides. This helps in discarding non-valid or extraneous solutions, which are common in radical equations.
• Iterative Approach: sometimes, multiple squaring or isolating steps are necessary, especially when handling complex or multiple radical terms.

Together, these methods form the backbone of efficiently tackling equations with radical elements, transforming them into straightforward algebraic problems.