Problem 3 - Intermediate Algebra

Question

For Problems 1-56, solve each equation. Don't forget to check each of your potential solutions. $$ \sqrt{2 x}+4=0 $$

Step-by-Step Solution

Answer

No real solution exists because the square root cannot be negative.

Step 1: Isolate the Radical Expression

Start by isolating the term with the square root. The equation is \( \sqrt{2x} + 4 = 0 \). Subtract 4 from both sides: \( \sqrt{2x} = -4 \).

Step 2: Analyze the Square Root Property

Remember that the square root of any real number, \( \sqrt{a} \), is always non-negative. Therefore, \( \sqrt{2x} = -4 \) is impossible for real numbers because a square root cannot equal a negative value.

Step 3: Conclude No Real Solution Exists

Since a square root cannot be negative, the equation \( \sqrt{2x} + 4 = 0 \) has no solutions among the real numbers. Thus, there is no value of \( x \) that satisfies the original equation.

Step 4: Verify and Validate

It's important to validate our conclusion by considering if there are any assumptions or alternative approaches. Given the nature of square roots in real numbers, no other real number solution is possible.

Key Concepts

Radical Expressions

Radical expressions are expressions that contain a root, such as a square root, cube root, etc. They are marked by the radical symbol \( \sqrt{} \). In math, square roots are the most common form of radical expressions you will encounter. Working with these expressions requires understanding how to manipulate them to simplify calculations.

• Radical expressions often consist of an expression under the radical sign, known as the radicand. For example, in \( \sqrt{2x} \), \( 2x \) is the radicand.
• The index of the radical determines the type of root, such as 2 for square roots (often not written) or 3 for cube roots.

Simplifying radicals often involves extracting factors that are perfect squares if dealing with square roots. Simplifying means expressing the radical in its simplest form where no radicands have perfect square factors other than 1.
Managing radical expressions is essential when solving equations, particularly when trying to isolate terms with roots, as showcased in the example equation \( \sqrt{2x} + 4 = 0 \).

Real Numbers

Real numbers include all numbers that can exist on the number line, encompassing positive and negative integers, fractions, decimals, and irrational numbers. They form a continuous set over the number line that you can use in everyday arithmetic operations.

• These numbers can either be rational (like \( \frac{3}{4} \), which can be expressed as a fraction) or irrational (like \( \sqrt{2} \), which cannot be neatly written as a simple fraction).
• Understanding real numbers is crucial because, when solving equations, it's necessary to know whether any solution exists within the "real world." Square roots of negative numbers, for example, are not considered real.

This is particularly relevant to the example equation \( \sqrt{2x} = -4 \), which was found to have no solution in the realm of real numbers, as square roots cannot yield negative results within this number set.

Square Root Property

The square root property deals with how we understand and manipulate square roots in equations. A fundamental characteristic of square root expressions is that their results are non-negative in the real number system.

• This means expressions like \( \sqrt{a} \) result in a non-negative number if \( a \) is a real number, something that comes into play when verifying the possibility of solutions in equations.
• In the step-by-step solution of the given equation, applying the square root property helped in recognizing that \( \sqrt{2x} = -4 \) was impossible for real numbers.

It's crucial for students to grasp the square root property's limitation to non-negative results. Whenever you are solving equations involving square roots, always check if any potential solution respects this property to confirm its validity.