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Mathematics is a subject that often intimidates students, but it is also a subject that can be incredibly empowering. One of the fundamental concepts in algebra is the formula of (a – b)². This formula allows us to simplify and solve complex equations, making it an essential tool for problemsolving. In this article, we will explore the formula of (a – b)² in detail, understand its derivation, and learn how to apply it effectively in various scenarios.
What is the Formula of (a – b)²?
The formula of (a – b)² is a mathematical expression that represents the square of the difference between two numbers, a and b. It can be expanded as follows:
(a – b)² = a² – 2ab + b²
This formula is derived from the concept of expanding binomial expressions. By squaring the difference between two numbers, we obtain a quadratic expression that consists of three terms: the square of the first number, twice the product of the two numbers, and the square of the second number.
Understanding the Derivation of the Formula
To understand the derivation of the formula of (a – b)², let’s consider a simple example:
(3 – 2)²
We can expand this expression by multiplying (3 – 2) with itself:
(3 – 2)² = (3 – 2) * (3 – 2)
Using the distributive property, we can expand this further:
(3 – 2) * (3 – 2) = 3 * (3 – 2) – 2 * (3 – 2)
Simplifying the multiplication:
3 * (3 – 2) – 2 * (3 – 2) = 3 * 3 – 3 * 2 – 2 * 3 + 2 * 2
Finally, simplifying the expression:
3 * 3 – 3 * 2 – 2 * 3 + 2 * 2 = 9 – 6 – 6 + 4 = 1
Therefore, (3 – 2)² = 1.
By following this process, we can derive the general formula of (a – b)² as a² – 2ab + b².
Applying the Formula of (a – b)²
The formula of (a – b)² has numerous applications in mathematics and reallife scenarios. Let’s explore some of the common applications:
1. Algebraic Simplification
The formula of (a – b)² allows us to simplify complex algebraic expressions. By expanding the expression using the formula, we can eliminate parentheses and combine like terms, making the equation easier to solve.
For example, consider the expression (x – 3)²:
(x – 3)² = x² – 2 * x * 3 + 3²
Simplifying further:
x² – 2 * x * 3 + 3² = x² – 6x + 9
By applying the formula, we have simplified the expression (x – 3)² to x² – 6x + 9.
2. Geometry
The formula of (a – b)² is also applicable in geometry, particularly in calculating areas and perimeters of squares and rectangles.
For example, consider a square with side length (a – b). The area of this square can be calculated using the formula (a – b)²:
Area = (a – b)²
Similarly, the perimeter of the square can be calculated by multiplying the side length by 4:
Perimeter = 4 * (a – b)
By applying the formula, we can easily calculate the area and perimeter of squares and rectangles.
3. Physics
The formula of (a – b)² is also relevant in physics, particularly in the study of motion and energy.
For example, consider the equation for kinetic energy:
Kinetic Energy = 0.5 * m * v²
where m represents mass and v represents velocity.
If we have two objects with masses (a – b) and velocities (a + b), we can calculate their kinetic energies using the formula of (a – b)²:
Kinetic Energy = 0.5 * (a – b) * (a + b)²
By applying the formula, we can determine the kinetic energies of the objects.
Examples and Case Studies
Let’s explore some examples and case studies to further illustrate the application of the formula of (a – b)²:
Example 1: Algebraic Simplification
Consider the expression (2x – 3y)². To simplify this expression, we can apply the formula of (a – b)²:
(2x – 3y)² = (2x)² – 2 * (2x) * (3y) + (3y)²
Simplifying further:
(2x)² – 2 * (2x) * (3y) + (3y)² = 4x² – 12xy + 9y²
Therefore, (2x – 3y)² simplifies to 4x² – 12xy + 9y².
Example 2: Geometry
Consider a rectangle with length (a + b) and width (a – b). To calculate the area of this rectangle, we can apply the formula of (a – b)²:
Area = (a + b) * (a – b)²
Simplifying further:
Area = (a + b) * (a² – 2ab + b²)
Expanding the multiplication:
Area = a * a² – a * 2ab + a * b² + b * a² – b * 2ab + b * b²
Simplifying the expression:
Area = a³ – 2a²b + ab² + ba² – 2ab² + b³
Combining like terms:
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