3-formula-solution.jpeg "(a-b)3 Formula Solution")
Understanding the (a-b)³ Formula
The formula (a-b)³ is a common algebraic expression used to expand the cube of a binomial. It represents the product of (a-b) multiplied by itself three times: (a-b) * (a-b) * (a-b). This formula is essential for simplifying expressions and solving various mathematical problems.
Expanding the Formula
The expanded form of (a-b)³ is:
a³ - 3a²b + 3ab² - b³
Let's break down how this formula is derived:
- First Expansion: (a-b) * (a-b) = a² - 2ab + b²
- Second Expansion: (a² - 2ab + b²) * (a-b) = a³ - 3a²b + 3ab² - b³
Applying the Formula
You can use the (a-b)³ formula to simplify expressions and solve equations. Here are some examples:
Example 1: Simplify (x-2)³
Using the formula, we get:
(x-2)³ = x³ - 3(x²)(2) + 3(x)(2²) - 2³ = x³ - 6x² + 12x - 8
Example 2: Solve for 'x' in the equation (x-1)³ = 8
- Expand the left side: x³ - 3x² + 3x - 1 = 8
- Move all terms to one side: x³ - 3x² + 3x - 9 = 0
- Factor the equation: (x-3)(x² + 3) = 0
- Solve for x: x = 3 or x² = -3. Since the square of a real number cannot be negative, the only solution is x = 3.
Remembering the Formula
The (a-b)³ formula can be easily remembered by following this pattern:
- First term: Cube of the first term (a³)
- Second term: Three times the product of the square of the first term and the second term (3a²b)
- Third term: Three times the product of the first term and the square of the second term (3ab²)
- Fourth term: Cube of the second term (b³)
Important Note: The signs alternate between positive and negative in the expanded form of the formula.
By understanding and applying the (a-b)³ formula, you can simplify expressions, solve equations, and enhance your understanding of algebraic manipulations.