yes why x square plus y square is equal to x plus y the hole square minus 2xy

The identity \( x^2 + y^2 = (x + y)^2 - 2xy \) comes from the expansion of the binomial square \((x + y)^2\). Let's break it down step by step.

### Step 1: Expand \((x + y)^2\)
We start with the expression for \((x + y)^2\):

\[
(x + y)^2 = (x + y) \times (x + y)
\]

Now, apply the distributive property:

\[
(x + y)^2 = x \times x + x \times y + y \times x + y \times y
\]

This simplifies to:

\[
(x + y)^2 = x^2 + xy + yx + y^2
\]

Since \( xy = yx \), the expression can be rewritten as:

\[
(x + y)^2 = x^2 + 2xy + y^2
\]

### Step 2: Isolate \( x^2 + y^2 \)
To isolate \( x^2 + y^2 \), subtract \( 2xy \) from both sides:

\[
x^2 + y^2 = (x + y)^2 - 2xy
\]

This equation is the identity that shows how \( x^2 + y^2 \) relates to \( (x + y)^2 \) and the product \( xy \).

### Conclusion
The identity \( x^2 + y^2 = (x + y)^2 - 2xy \) is a result of expanding and rearranging the square of the sum \( (x + y)^2 \). It allows us to express the sum of the squares of two numbers in terms of their sum and product.

Do you have any further questions or need more clarification?

### Related Questions
1. How can this identity be used in solving quadratic equations?
2. What are other useful identities involving squares of sums or differences?
3. How does the identity change if we consider three numbers instead of two?
4. Can you derive the identity for the difference of squares?
5. How does this identity help in vector algebra?

### Tip
Knowing and understanding algebraic identities, such as the square of a sum or difference, is crucial for simplifying complex expressions and solving equations efficiently.

Explore the algebraic identity x^2 + y^2 = (x + y)^2 - 2xy with a detailed explanation and step-by-step derivation. This identity arises from expanding the binomial square (x + y)^2 and is essential in algebraic simplification and equation solving. Suitable for high school students.

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