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Chapter 5: Problem 3

Solve each equation. Don't forget to check each of your potential solutions. \(\sqrt{2 x}+4=0\)

Short Answer

Expert verified
No real solution exists because a square root cannot be negative.

Step by step solution

01

Isolate the Square Root

We begin by isolating the square root term in the equation. Our equation is \( \sqrt{2x} + 4 = 0 \). To isolate the square root, subtract 4 from both sides: \( \sqrt{2x} = -4 \).
02

Examine the Equation for Real Solutions

Now, consider the equation \( \sqrt{2x} = -4 \). The square root of any real number cannot be negative, so \( \sqrt{2x} = -4 \) has no real solutions. Therefore, we conclude that there are no real solutions for the equation \( \sqrt{2x} + 4 = 0 \) because no real number squared can give a negative result.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Square Root Equations
When dealing with square root equations, we often encounter expressions like \( \sqrt{2x} \). The major step in solving these equations is to remove or 'isolate' the square root from the rest of the equation. This helps us focus on the main part where the calculation is needed.
  • Isolating the Square Root: To solve \(\sqrt{2x} + 4 = 0\), you should first get the square root term, \(\sqrt{2x}\), by itself. This can be done by subtracting 4 from both sides, leading to \(\sqrt{2x} = -4\).
  • Remember, when you isolate the square root, you're trying to simplify the problem to a basic form that can be more readily tackled.
  • Once isolated, the next step usually involves removing the square root itself, often by squaring both sides of the equation. In cases where the other side of the equation is negative, as here, it鈥檚 crucial to consider the implications for real number solutions.
The main takeaway here is understanding the process of isolating the square root: it's a systematic way to break down complex equations into simpler parts.
Real Number Solutions
Understanding real number solutions is crucial in solving equations, especially when square roots are involved. A real number is any number that can be found on the number line, including both positive and negative numbers, and zero. However, square roots offer a unique twist.
The square root of a number, \( \sqrt{x} \), represents a value that, when multiplied by itself, gives \(x\). Importantly, the square root of a positive number is always positive (or occasionally zero).
  • One key rule is that square roots of real numbers cannot have negative values. For instance, \( \sqrt{2x} = -4\) is impossible because square roots yield non-negative outcomes only.
  • If an equation results in a statement like \( \sqrt{2x} = -4\), one can quickly conclude there are no real solutions, as no real number squared equals a negative number.
Understanding these principles helps recognize situations where no real number solutions exist.
Equation Isolation
Equation isolation is the process of rearranging the equation to focus on the part of the equation that makes it "solvable." The ultimate goal is to separate the variable in question from the rest of the terms.
In the exercise \( \sqrt{2x} + 4 = 0\), you isolate \(\sqrt{2x}\) by subtracting 4 from both sides, resulting in \(\sqrt{2x} = -4\). This manipulation allows a clearer view of the relationship between the variable \(x\) and its determining values.
  • First, always perform operations that maintain the equation's balance, meaning whatever you do to one side of the equation, you must do to the other.
  • By isolating components, especially complex ones like square roots, you simplify the equation's structure, making it easier to analyze and solve.
  • Sometimes, isolation reveals that real solutions are not possible, as is the case in our example, where the isolated square root equation yields no real solutions due to the impossible condition of a negative square root.
Mastering the technique of equation isolation is fundamental in tackling and solving more complex mathematical problems efficiently.

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Most popular questions from this chapter

For Problems \(19-32\), write each of the following in ordinary decimal notation. For example, \((3.18)(10)^{2}=318\). \((7.631)(10)^{4}\)

Perform the indicated operations and express answers in simplest radical form. (See Example 5.) \(\frac{\sqrt[3]{3}}{\sqrt[4]{3}}\)

For Problems \(19-32\), write each of the following in ordinary decimal notation. For example, \((3.18)(10)^{2}=318\). \((2.04)(10)^{12}\)

Simplify each of the following. Express final results using positive exponents only. For example,\(\left(2 x^{\frac{1}{2}}\right)\left(3 x^{\frac{1}{3}}\right)=6 x^{\frac{5}{6}}\). \(\left(\frac{2 x^{\frac{1}{3}}}{3 y^{\frac{1}{4}}}\right)^{4}\)

Use scientific notation and the properties of exponents to help you perform the following operations. \((0.00007)(11,000)\)
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