Solving Equations: Finding the Value of 2a When 4a3 5a - 2

This article aims to illustrate the process of solving an algebraic equation to find the value of 2a, specifically when the equation is given as 4a3 5a - 2. We will go through several methods to solve the equation step-by-step and derive the value of a, and subsequently, 2a. The methods discussed will include grouping like terms, subtracting and adding terms, and simplifying expressions.

Solving 4a3 5a - 2 to Find the Value of 2a

To find the value of 2a, we first need to determine the value of a from the equation 4a3 5a - 2.

Method 1: Grouping Like Terms

Begin by rearranging the terms to group like terms on one side of the equation:

4a3 5a - 2

4a3 - 5a -2

To find the value of a, let's manipulate the equation to isolate a. However, the given equation involves a cubic term (4a3), which makes it more complex than a linear equation. Let's first simplify and see if we can simplify the equation by combining terms involving a:

4a3 - 5a -2

4a3 - 5a -2

Since 4a3 and 5a are not like terms, grouping them directly is not straightforward. Instead, let's check if there is a simpler way to solve this equation by isolating the term involving a:

4a3 - 5a 2 0

Now, solving this cubic equation requires more advanced techniques such as factoring or numerical methods, but it is important to note that a simpler approach might not be immediately apparent.

Method 2: Adding and Subtracting Terms

Let's try to manipulate the equation by adding and subtracting terms to simplify the equation:

4a3 5a - 2

Subtract 4a2 from both sides:

4a3 - 4a2 5a - 2 - 4a2

Factor out a on the left side:

4a2(a - 1) 5a - 2 - 4a2

This is not a straightforward simplification. Instead, let's go back to the original equation and try to isolate a:

4a3 - 5a -2

3 - 2 5a - 4a

a 1

Now that we have a 1, we can find the value of 2a:

2a 2(1) 2

Method 3: Simplifying Using Arithmetic Operations

Another approach is to simplify the equation by performing arithmetic operations:

4a3 - 5a -2

3 - 2 5a - 4a

Since 3 - 2 1 and 5a - 4a a:

a 1

Now, we can find the value of 2a:

2a 2(1) 2

Method 4: Simplifying the Expression

A fourth way to solve this involves directly manipulating the equation to simplify and find the value of a:

4a3 5a - 2

3 - 2 5a - 4a

Since the equation becomes:

5 2a

Multiply both sides by 2 to get:

2a 10

Hence, the value of 2a is 10.

Conclusion

By exploring different methods, we have determined that the value of 2a in the equation 4a3 5a - 2 is 10. The process involved manipulating the original equation, isolating the variable, and then performing the necessary arithmetic to find the value. This problem highlights the importance of careful algebraic manipulation and the ability to identify simplifying steps to solve more complex equations.