(5x-3).jpeg "(5x+3)(5x-3)")
Understanding the Special Product (5x + 3)(5x - 3)
The expression (5x + 3)(5x - 3) represents a special product in algebra, often referred to as the difference of squares. This pattern arises because the two binomials are identical except for the sign between their terms.
The Difference of Squares Pattern
The general form of the difference of squares is:
(a + b)(a - b) = a² - b²
This pattern shows that when you multiply two binomials with the same terms but opposite signs, the result is the square of the first term minus the square of the second term.
Applying the Pattern to (5x + 3)(5x - 3)
Let's apply this pattern to our expression:
- Identify 'a' and 'b': In this case, a = 5x and b = 3.
- Square 'a' and 'b': a² = (5x)² = 25x² and b² = 3² = 9
- Apply the pattern: (5x + 3)(5x - 3) = a² - b² = 25x² - 9
Why This Pattern Matters
Understanding the difference of squares pattern is crucial because it allows us to:
- Simplify expressions quickly: Instead of manually multiplying out the binomials, we can directly apply the pattern to obtain the simplified form.
- Factorize expressions easily: Recognizing the pattern helps us factorize expressions that follow the difference of squares form.
- Solve equations more efficiently: This pattern can be used to factorize quadratic equations, leading to simpler solutions.
Example:
Let's say we want to find the value of (5x + 3)(5x - 3) when x = 2.
Instead of substituting x = 2 directly into the original expression, we can use the difference of squares pattern:
(5x + 3)(5x - 3) = 25x² - 9
Substituting x = 2:
25(2)² - 9 = 25(4) - 9 = 100 - 9 = 91
By applying the pattern, we simplified the calculation and obtained the answer much faster.