Question

\(\sqrt{5 x}+5=0\)

Short answer

The equation has no real solutions. A square root of a number can't be a negative number.

Step-by-step solution

Isolating the square root term

Firstly, isolate the square root term on one side of the equation by removing the number 5 from it. This can be done by subtracting 5 from both sides of the equation.So, the equation will look like this: \(\sqrt{5x} = -5\).

Checking for non-real solutions

As the square root of a number cannot be negative, this equation has no solution in the set of real numbers. Nothing squared can be a negative number.

No Solution

Since we can't obtain a real number solution for the equation, we stop at this step and conclude the equation has no real solutions.

Key concepts

Real Numbers

Real numbers are fundamental in math and include all the numbers on the number line. They can be classified into many types:

• Rational numbers: These are numbers that can be expressed as a ratio of two integers (e.g., 1/2 or -3).
• Irrational numbers: Numbers that cannot be expressed as a simple fraction (e.g., \( \pi \) or \( \sqrt{2} \)).
• Whole numbers: These are non-negative integers, including zero.
• Integers: These include whole numbers and their negative counterparts (e.g., -1, -2, 3, etc.).
• Natural numbers: These are positive integers, starting from 1.

Real numbers are important because they cover almost all the numbers used in everyday life. They form a continuous line, without gaps, which allows us to solve many types of algebraic and calculus problems. When solving equations like \(\sqrt{5x} + 5 = 0\), it's crucial to determine if solutions lie within the real number set.

Isolating Square Roots

When solving equations involving square roots, the first step often involves isolating the radical term. This means getting the \(\sqrt{}\) expression by itself on one side of the equation. In the exercise \(\sqrt{5x} + 5 = 0\), we isolate the square root by subtracting 5 from both sides, resulting in \(\sqrt{5x} = -5\).

The process of isolating the square root is critical as it sets the stage for finding potential solutions to the equation. Once isolated, we would usually square both sides to eliminate the square root, but caution is needed. If a negative number is isolated inside a square root, as in this exercise, it indicates the absence of real solutions. The idea is to check whether squaring would yield a valid real number.

Non-real Solutions

In mathematical equations, non-real solutions refer to answers that do not belong to the set of real numbers. Real numbers cover numbers we can represent on the number line, while non-real, or imaginary numbers, include complex components.

For the exercise \(\sqrt{5x} = -5\), an attempt to find a real solution highlights this concept. The square root of any real number is always non-negative, meaning the isolated \(\sqrt{5x}\) cannot equal -5. This reveals that the equation yields no real solution.

Understanding when equations have no real solutions helps prevent unnecessary calculations. It also reinforces comprehension of number types and the conditions under which various operations are valid. Thorough checking of whether an equation leads to a negative square root is a handy skill in identifying non-real solutions early in your problem-solving process.