Problem 93 - Elementary and Intermediate Algebra

Question

Solve each equation. Write all proposed solutions. Cross out those that are extraneous. $$ \sqrt{2 x}+5=1 $$

Step-by-Step Solution

Answer

No real solutions.

Step 1: Isolate the Radicals

The given equation is \( \sqrt{2x} + 5 = 1 \). First, isolate the radical term by subtracting 5 from both sides of the equation.\[ \sqrt{2x} = 1 - 5 \]\[ \sqrt{2x} = -4 \]

Step 2: Analyze the Equation

At this stage, we must consider whether the equation has a real solution. Recall that the square root of a number cannot be negative in the set of real numbers. Hence, \(\sqrt{2x} = -4\) leads to a contradiction.

Step 3: Conclusion on Solutions

The equation \( \sqrt{2x} = -4 \) implies there are no real solutions because a square root cannot be a negative number. Thus, the original equation has no real solutions.

Key Concepts

Isolating Radicals

When solving an equation with a radical, such as a square root, the first step is to isolate this expression. **Why is this important?** Isolating the radical helps us to better manipulate and understand the equation.

Here's how you can do it:

• Identify the term with the radical. For the equation \( \sqrt{2x} + 5 = 1 \), the term \( \sqrt{2x} \) is what we need to isolate.
• Use inverse operations to move other terms to the other side of the equation. In this example, subtract 5 from both sides, which results in \( \sqrt{2x} = -4 \).

These steps allow us to clearly see any contradictions or further transformations needed for a complete solution. **Remember:** Isolating radical terms is a critical step before squaring both sides or examining the equation for solutions.

Analyzing Equations

Once a radical is isolated, the next step is analyzing the equation to determine possible solutions. **How do we do this?** We need to check the properties of radicals.

• A square root always returns a non-negative output. This is because square roots are defined in the real numbers realm as non-negative. For example, \( \sqrt{x} \geq 0 \) for any real number \( x \geq 0 \).
• After isolating, if you find the equation states something like \( \sqrt{2x} = -4 \), pause to consider its meaning. Such a statement contradicts the principal definition of square roots.

Recognizing contradictions is a part of analyzing. It helps us conclude whether a real solution exists or if the expression leads us to an invalid scenario, showcasing the need for critical examination during problem solving.

Extraneous Solutions

**What are extraneous solutions?** These are solutions derived from solving an equation that do not satisfy the original equation. They often appear when we manipulate equations algebraically, particularly through squaring both sides.

• When we square both sides of an equation, new solutions may appear that are not valid for the original equation.
• In the given exercise, after isolating to \( \sqrt{2x} = -4 \), this term alone shows contradiction without the need for further algebraic manipulation. Yet, typically, if we continue by squaring, we'd arrive at \( 2x = 16 \), but solving it doesn’t work as there's no real \( x \) satisfying the square root equation.

It's crucial to always test or check solutions with the original equation. This ensures that every found solution is correct and not extraneous. Validating solutions keeps our problem-solving process clear and accurate.