(3a-4b).jpeg "(3a+4b)(3a-4b)")
Understanding (3a+4b)(3a-4b)
The expression (3a+4b)(3a-4b) is a product of two binomials. It can be solved using the FOIL method or by recognizing it as a special case known as the difference of squares.
Using the FOIL Method
FOIL stands for First, Outer, Inner, Last. This method helps us multiply the terms of the binomials systematically:
- First: Multiply the first terms of each binomial: (3a) * (3a) = 9a²
- Outer: Multiply the outer terms: (3a) * (-4b) = -12ab
- Inner: Multiply the inner terms: (4b) * (3a) = 12ab
- Last: Multiply the last terms: (4b) * (-4b) = -16b²
Now, add all the terms together: 9a² - 12ab + 12ab - 16b²
Simplifying, we get: 9a² - 16b²
Recognizing the Difference of Squares
The expression (3a+4b)(3a-4b) fits the pattern of the difference of squares:
(a + b)(a - b) = a² - b²
In our case, 'a' is represented by 3a and 'b' by 4b. Applying this pattern directly, we get:
(3a)² - (4b)² = 9a² - 16b²
Conclusion
Both methods lead to the same answer: 9a² - 16b². The difference of squares pattern offers a quicker and more elegant way to solve this type of problem, especially as you become more familiar with it.