(2/3x+4)(3/2x+6)-(1/7x-1)(1/7x+1)

3 min read Jun 16, 2024

Simplifying Algebraic Expressions: (2/3x+4)(3/2x+6)-(1/7x-1)(1/7x+1)

This article will guide you through the process of simplifying the algebraic expression: (2/3x+4)(3/2x+6)-(1/7x-1)(1/7x+1).

Step 1: Expanding the Products

We begin by expanding the products using the FOIL method (First, Outer, Inner, Last).

(2/3x+4)(3/2x+6)

First: (2/3x) * (3/2x) = x²
Outer: (2/3x) * 6 = 4x
Inner: 4 * (3/2x) = 6x
Last: 4 * 6 = 24

Therefore, (2/3x+4)(3/2x+6) = x² + 4x + 6x + 24 = x² + 10x + 24

(1/7x-1)(1/7x+1)

First: (1/7x) * (1/7x) = 1/49x²
Outer: (1/7x) * 1 = 1/7x
Inner: -1 * (1/7x) = -1/7x
Last: -1 * 1 = -1

Therefore, (1/7x-1)(1/7x+1) = 1/49x² + 1/7x - 1/7x - 1 = 1/49x² - 1

Step 2: Combining the Expanded Terms

Now we substitute the expanded forms back into the original expression:

(2/3x+4)(3/2x+6)-(1/7x-1)(1/7x+1) = (x² + 10x + 24) - (1/49x² - 1)

Step 3: Simplifying the Expression

Finally, we simplify the expression by combining like terms:

x² terms: x² - 1/49x² = (49/49)x² - (1/49)x² = 48/49x²
x terms: 10x
Constant terms: 24 + 1 = 25

Therefore, the simplified expression is: 48/49x² + 10x + 25

Conclusion

By following these steps, we have successfully simplified the complex algebraic expression: (2/3x+4)(3/2x+6)-(1/7x-1)(1/7x+1) into a more manageable form: 48/49x² + 10x + 25. This process illustrates the importance of understanding basic algebraic operations like expansion and combining like terms.