%5E2-formula-proof.jpeg "(x-y-z)^2 Formula Proof")
The Proof of (x-y-z)^2 Formula
The formula for $(x-y-z)^2$ is a fundamental concept in algebra, and it is widely used in various mathematical expressions. In this article, we will provide a step-by-step proof of the formula.
The Formula
The formula for $(x-y-z)^2$ is given by:
$(x-y-z)^2 = x^2 - 2xy + y^2 - 2xz + 2yz - z^2$
The Proof
To prove this formula, we can start by expanding the expression $(x-y-z)^2$ using the distributive property of multiplication over addition.
$(x-y-z)^2 = (x-y-z)(x-y-z)$
Now, we can multiply the two binomials:
$(x-y-z)(x-y-z) = x^2 - xy - xz - yx + y^2 + yz + zx - zy - z^2$
Next, we can combine like terms:
$x^2 - xy - xz - yx + y^2 + yz + zx - zy - z^2$ $= x^2 - 2xy + y^2 - 2xz + 2yz - z^2$
Thus, we have derived the formula for $(x-y-z)^2$.
Example
Let's consider an example to illustrate the formula. Suppose we want to expand $(2-3-4)^2$. Using the formula, we get:
$(2-3-4)^2 = 2^2 - 2(2)(3) + 3^2 - 2(2)(4) + 2(3)(4) - 4^2$ $= 4 - 12 + 9 - 16 + 24 - 16$ $= -7$
Therefore, $(2-3-4)^2 = -7$.
Conclusion
In conclusion, the formula for $(x-y-z)^2$ is a powerful tool that enables us to expand expressions involving the square of a binomial. The proof of the formula involves a simple application of the distributive property of multiplication over addition, and it can be used to simplify complex algebraic expressions.
