Problem 405 - Elementary Algebra

Question

In the following exercises, solve. \(\sqrt{5 v-2}+5=0\)

Step-by-Step Solution

Answer

No real solution.

Step 1: Isolate the Square Root Term

Start by isolating the square root term on one side of the equation. Subtract 5 from both sides to get: ewline ewline \(ewline ewline \sqrt{5v - 2} = -5\)ewline

Step 2: Analyze the Equation

Notice that the square root of any real number cannot be negative. Therefore: \(ewline ewline \sqrt{5v - 2} = -5\) has no real solution.ewline

Key Concepts

Isolating Terms

To solve equations involving square roots, it’s essential to isolate the square root term. This means you need to have the square root on one side of the equation and everything else on the other.

In our example, the equation \(ewline\ \sqrt{5 v - 2}+5=0\)\ starts by having a +5 on the same side as the square root term. To isolate the square root, we subtract 5 from both sides. This gives us:

\[\ \sqrt{5v - 2} = -5\ \]

Now, the square root term \(ewline\ \sqrt{5v - 2}\ \)\ stands alone on one side (left) of the equation.

Non-Negative Property of Square Roots

One fundamental property of square roots is that they cannot be negative. The square root of any real number will always return a non-negative result.

In our isolated equation:

\(\ \sqrt{5v - 2} = -5\ \)

This equation suggests that the square root of \(ewline \ 5v - 2 \)\ is equal to -5, which is not possible. Given the non-negative nature of square roots, such an expression has no real (math) solution. A negative number on the right side invalidates the equation.

It’s important to check the possible outcomes after isolating terms to determine if the equation has a feasible solution or not.

Analyzing Equations

Analyzing an equation involves understanding its properties and implications. After isolating the square root term and noting the non-negative property, it's clear our equation has no real solution.

When you see an equation like this, always:
* Isolate the square root term
* Check the properties of the square root, particularly it being non-negative

These steps ensure you are not trying to solve an impossible problem. When you find that an equation suggests the square root of a number equals a negative, it means there’s no solution within the realm of real numbers.

It’s also a good practice to go back to your initial set-up to confirm the isolated term was correctly derived. This helps avoid mistakes and reinforces understanding of key mathematical principles.