%5E2-formula-proof.jpeg "(x+y+z)^2 Formula Proof")
The Formula Proof of (x + y + z)^2
The expansion of (x + y + z)^2 is a fundamental concept in algebra, and it's essential to understand its proof to appreciate the beauty of mathematics. In this article, we'll delve into the step-by-step proof of the formula.
The Formula
The expansion of (x + y + z)^2 is given by:
(x + y + z)^2 = x^2 + y^2 + z^2 + 2xy + 2xz + 2yz
The Proof
To prove this formula, we'll start by using the distributive property of multiplication over addition, which states that:
(a + b)(c + d) = ac + ad + bc + bd
Now, let's apply this property to (x + y + z)^2:
(x + y + z)^2 = (x + y + z)(x + y + z)
Step 1: Expand the Right-Hand Side
Using the distributive property, we can expand the right-hand side as follows:
(x + y + z)(x + y + z) = x(x + y + z) + y(x + y + z) + z(x + y + z)
Step 2: Expand Each Term
Now, let's expand each term:
x(x + y + z) = x^2 + xy + xz y(x + y + z) = xy + y^2 + yz z(x + y + z) = xz + yz + z^2
Step 3: Combine Like Terms
Combine the like terms:
x^2 + xy + xz + xy + y^2 + yz + xz + yz + z^2
Step 4: Simplify the Expression
Simplify the expression by combining the like terms:
x^2 + y^2 + z^2 + 2xy + 2xz + 2yz
And that's it! We've successfully proven the formula for (x + y + z)^2.
Conclusion
The proof of (x + y + z)^2 is a beautiful example of how algebraic manipulations can lead to a profound formula. This formula has numerous applications in various fields, including physics, engineering, and computer science. By understanding the proof of this formula, you'll gain a deeper appreciation for the intricacies of mathematics.
