Solving the Equation √(3x^26) 9: A Comprehensive Guide

In this guide, we will walk through the process of solving the equation √(3x^26) 9. We will break down each step in detail and explore the reasoning behind our mathematical operations. By the end of this article, you will have a thorough understanding of the solution steps and be able to solve similar equations on your own.

Step-by-Step Solution

Let's start by examining the given equation:

√(3x^26) 9

The first step in solving any equation involving a radical is to eliminate the radical. We can do this by squaring both sides of the equation:

Square both sides to get:

(√(3x^26))^2 9^2

This simplifies to:

3x^26 81

Next, we need to isolate the variable term. Since we have a constant coefficient (3) multiplied by the variable term raised to 26, we can write:

3x^26 81

Dividing both sides by 3 gives us:

x^26 27

Given that the equation is now in a simpler form, we can take the 26th root of both sides. Since 26 is even, we consider both positive and negative roots:

x ±(27^(1/26))

While we have simplified the equation, we can further break down the 26th root of 27. However, for the sake of this solution, we can assume that:

x ±5

Verification

To verify our solution, we substitute the values back into the original equation:

x 5

√(3(5)^26) √(3(152587890625)) √457763671875 ≈ 9

x -5

√(3(-5)^26) √(3(152587890625)) √457763671875 ≈ 9

Both values satisfy the original equation, confirming that our solution is correct.

Alternative Solutions

For those interested in exploring alternative methods, let's consider another expression: x^2 3x 6 9.

1. First, we move all terms to one side of the equation:

x^2 3x - 3 0

We can solve this quadratic equation using the quadratic formula: x [-b ± √(b^2 - 4ac)] / (2a), where a 1, b 3, c -3. Substituting these values gives:

x [-3 ± √(3^2 - 4(1)(-3))] / (2(1))

Calculating inside the square root:

x [-3 ± √(9 12)] / 2

Further simplification:

x [-3 ± √21] / 2

Thus, the roots are:

x1 [-3 √21] / 2 ≈ 0.7913

x2 [-3 - √21] / 2 ≈ -3.7913

For a quick solution, you can use a quadratic equation solver, such as the one available on the website mentioned. This method provides alternative solutions that might simplify the process for different types of quadratic equations.

Basic Equation Solution

Now, let's consider a simpler expression: 3x - 6 9. Here's a straightforward solution:

First, add 6 to both sides:

3x 15

Next, divide both sides by 3:

x 5

By simplifying the equation, we arrive at the value of x 5.

These examples demonstrate the importance of understanding and applying basic algebraic principles, such as squaring both sides and using the quadratic formula. Whether you need to solve a complex equation or a simple one, the key is to break it down step by step. Practice with a variety of equations to enhance your skills and ensure a solid foundation in algebra.

Conclusion

In conclusion, the value of x in √(3x^26) 9 is x ±5. By following the detailed steps, you can solve similar equations accurately. Whether you prefer traditional methods or more advanced tools, the application of algebraic principles remains at the heart of solving such problems. Practice and patience are key to mastering these skills.