Solving the Equation 321x - 3x 3x3 - 3

In this article, we will solve the equation 321x - 3x 3x3 - 3. We will break down the solution step by step, using mathematical concepts such as the order of operations (BODMAS) and algebraic techniques for solving equations.

Introduction to the BODMAS Rule

The BODMAS rule is an acronym used for remembering the order of operations in mathematics. It stands for Brackets, Orders, Division and Multiplication (from left to right), Addition and Subtraction (from left to right).

Solving the Equation Using the BODMAS Rule

To solve the equation 321x - 3x 3x3 - 3, we need to carefully apply the BODMAS rule and algebraic methods.

Step 1: Rewrite the Equation with Parentheses

Let's assume the equation should read 3(21x) - 3x (3x3 - 3).

First, we simplify the right side of the equation:

3(21x) - 3x 3(x3) - 3

Step 2: Simplify the Equation

Next, we simplify the left side and the right side of the equation:

3(21x) - 3x 3(x3) - 3

This simplifies to:

9x2 - 3x 27x - 3

Step 3: Introduce a Substitution Variable

Let n 3x. Substituting this into the equation, we get:

9n2 - n 27n - 3

Step 4: Rearrange the Equation

Rearrange the equation to form a standard quadratic equation:

9n2 - 28n 3 0

Step 5: Solve the Quadratic Equation

Using the quadratic formula, we solve for n:

n (28 ± √(282 - 493))/18

This simplifies to:

n (28 ± √(784 - 372))/18

Further simplification gives:

n (28 ± √412)/18

which can be simplified further to:

n (14 ± √103)/9

So, we get two values for n:

n 1/9 or n 3

Step 6: Solve for x

Since n 3x, we get two solutions for x:

3x 1/32 or 3x 31

This simplifies to:

3x 3-2 or x -2

And:

3x 31 or x 1

Conclusion

The solution set for the equation 321x - 3x 3x3 - 3 is {2, 1}

This demonstrates the process of solving an exponential equation using the BODMAS rule and algebraic methods. It is important to carefully apply the BODMAS rule and understand the concepts of orders and substitution for solving such equations.

Keywords: Solving exponential equations, Algebraic methods, BODMAS rule, Quadratic equations