## A-B Whole Square

The algebraic identity (๐โ๐)ยฒ represents the square of the difference between two numbers, ๐ and ๐. This formula expands to ๐ยฒโ2๐๐+๐ยฒ, integrating concepts from integers, rational numbers, and irrational numbers. It finds utility in various mathematical fields including algebra, where it helps simplify expressions and solve equations. The identity also plays a role in statistical methods like the least squares method, which is used for data fitting. Understanding (๐โ๐)ยฒ is fundamental in exploring more complex numerical and algebraic studies, including square and square roots.

## What is (a – b) Whole Square Formula?

The formula for (๐โ๐)ยฒ, commonly referred to as the square of a binomial difference, is an important algebraic identity. It is expressed as:

**(๐-๐)ยฒ = ๐ยฒ-2๐๐+๐ยฒ**

This formula represents the expanded form of squaring the difference between any two numbers, ๐ and ๐. Here’s a breakdown of the components:

- ๐ยฒ: the square of the first term.
- โ2๐๐: twice the product of the two terms, with a negative sign indicating subtraction.
- ๐ยฒ: the square of the second term.

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## Proof of A minus B Whole Square Formula

To prove the algebraic identity (๐-๐)ยฒ = ๐ยฒ-2๐๐+๐ยฒ, we can use the method of expanding the expression through basic algebraic principles. Here’s the step-by-step proof:

### Step 1: Write the expression

Start with the expression (๐โ๐)ยฒ. This signifies the square of the binomial ๐โ๐.

### Step 2: Expand the square

Remember that squaring a binomial involves multiplying the binomial by itself:

(๐โ๐)ยฒ = (๐โ๐)(๐โ๐)

### Step 3: Apply the distributive property (also known as the FOIL method for binomials)

**First terms**: Multiply the first term in each binomial: ๐โ ๐ = ๐2**Outer terms**: Multiply the outer terms of the binomials: ๐โ (โ๐) = โ๐๐**Inner terms**: Multiply the inner terms of the binomials: (โ๐)โ ๐ = โ๐๐**Last terms**: Multiply the last terms in each binomial:(โ๐)โ (โ๐) = ๐ยฒ

### Step 4: Combine like terms

Now, combine all the terms from the expansion:

๐ยฒโ๐๐โ๐๐+๐ยฒ

Combine the middle terms:

โ๐๐โ๐๐ = โ2๐๐

### Step 5: Write the final expression

So, the expression simplifies to:

๐ยฒโ2๐๐+๐ยฒ

This completes the proof that (๐-๐)ยฒ = ๐ยฒ-2๐๐+๐ยฒ. This identity is very useful in algebra for simplifying expressions and solving equations, and it holds true for all real numbers, including integers, rational numbers, and irrational numbers.

## Examples of A-B Whole Square

The formula for (aโb)ยฒ is a fundamental algebraic identity used to expand and simplify expressions. The identity is:

(aโb)ยฒ = abยฒโ2ab+bยฒ

This formula shows that the square of the difference between two terms, a and b, is the square of the first term, minus twice the product of the two terms, plus the square of the second term. Here are some examples to illustrate how to apply this formula in various scenarios:

## Example 1: Basic Numbers

**Problem:** Calculate (5โ3)ยฒ.

**Solution:** Using the formula:

(5โ3)ยฒ = 5ยฒโ2โ 5โ 3+3ยฒ = 25โ30+9 = 4

So, (5โ3)ยฒ = 4.

## Example 2: Algebraic Terms

**Problem:** Simplify (๐ฅโ4)2(*x*โ4)2.

**Solution:** Apply the formula:

(๐ฅโ4)ยฒ = ๐ฅยฒโ2โ ๐ฅโ 4+4ยฒ = ๐ฅ2โ8๐ฅ+16

Thus, (๐ฅโ4)ยฒ simplifies to ๐ฅยฒโ8๐ฅ+16.

## Example 3: Variables with Coefficients

**Problem:** Expand (3๐โ2๐)ยฒ.

**Solution:** Using the identity:

(3๐โ2๐)ยฒ = (3๐)ยฒโ2โ 3๐โ 2๐+(2๐)ยฒ = 9๐ยฒโ12๐๐+4๐ยฒ

So, (3๐โ2๐)ยฒ expands to 9๐ยฒโ12๐๐+4๐ยฒ.

## FAQs

## What are some practical applications of the (๐โ๐)ยฒ formula in real-life scenarios?

In real-life, the (๐โ๐)ยฒ formula can be used in project planning to calculate variances, in finance to compute financial forecasts and risk assessments, and in engineering to design and analyze the stability of structures. It also plays a role in optimizing processes and solving problems that involve squared differences.

## Why is the (๐โ๐)ยฒ formula important in mathematics?

The (๐โ๐)ยฒ formula is crucial for simplifying and solving algebraic equations, aiding in data analysis (e.g., in statistical methods like the least squares method), and understanding geometric relationships. It’s a foundational tool in algebra that extends to various applications in higher mathematics and applied sciences.

## What is the formula for (๐โ๐)ยฒ and what does each term represent?

The formula for (๐โ๐)ยฒ is ๐ยฒโ2๐๐+๐ยฒ. Here, ๐ยฒ represents the square of the first term, โ2๐๐ is twice the product of the two terms and indicates subtraction, and ๐ยฒ is the square of the second term. This identity helps simplify and solve quadratic expressions in algebra.