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Magazine Chapter 10: Problem 5
Solve. $$ \sqrt{2 x}=-4 $$
Short Answer
Expert verified
No real solution exists.
Step by step solution
01
Understand the Equation
The equation given is \( \sqrt{2x} = -4 \). It involves a square root operation on \( 2x \), which is set equal to a negative number.
02
Consider the Properties of Square Roots
A square root function, \( \sqrt{a} \), outputs a non-negative number for any input \( a \geq 0 \). Thus, \( \sqrt{a} \) can never equal a negative number, such as -4.
03
Analyze for Real Solutions
Since a square root cannot be negative, there is no real number \( x \) that will satisfy the equation \( \sqrt{2x} = -4 \).
04
Conclusion About the Solution
Since the condition \( \sqrt{2x} = -4 \) is impossible with real numbers, the equation has no solution in the set of real numbers.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Square Roots
Square roots are a fundamental concept in algebra and mathematics. The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3 because \(3 \times 3 = 9\). Square roots of positive numbers are always non-negative. This is why the result of a square root operation cannot be a negative number.
- The square root symbol is \(\sqrt{}\), and it typically produces a principal (non-negative) result.
- For example, \(\sqrt{4} = 2\) because \(2 \times 2 = 4\).
Properties of Exponents
Exponents describe how many times a number called the base is multiplied by itself. For instance, \(2^3\) means \(2 \times 2 \times 2\). Many rules govern the manipulation of exponents, and understanding these rules helps in simplifying and solving equations involving exponents.
- Multiplication: When multiplying powers with the same base, you add the exponents: \(a^m \cdot a^n = a^{m+n}\).
- Power of a power: When taking a power of a power, you multiply the exponents: \((a^m)^n = a^{m\cdot n}\).
- Division: When dividing powers with the same base, you subtract the exponents: \(a^m / a^n = a^{m-n}\).
Real Numbers
Real numbers encompass all the numbers we typically encounter in everyday life, including integers, fractions, and decimals. They can be rational, meaning they can be expressed as a fraction of two integers, or irrational, which cannot be expressed perfectly as a fraction.
- Rational Numbers: Examples include 1/2, 3, and -7.
- Irrational Numbers: Examples include \(\pi\) and \(\sqrt{2}\).
- Properties: Real numbers are closed under addition, subtraction, multiplication, and division (except by zero).
Solving Equations
Solving equations is a fundamental skill in algebra, involving finding the value(s) of the variable(s) that make an equation true. This usually involves manipulating the equation using a variety of algebraic techniques.
- Isolate the variable: Use operations like addition, subtraction, multiplication, and division to get the variable on one side of the equation.
- Check for extraneous solutions: Verify that the solutions satisfy the original equation.
- Consider the domain: Not all mathematical solutions are within the real numbers, so understand the context.
