%5E2-formula.jpeg "(x-y-z)^2 Formula")
The (x-y-z)^2 Formula: A Comprehensive Guide
The (x-y-z)^2 formula is a fundamental concept in algebra and is widely used in various mathematical operations. In this article, we will explore the formula, its expansion, and some examples to illustrate its application.
What is the (x-y-z)^2 Formula?
The (x-y-z)^2 formula is a quadratic expression that represents the square of the difference between three variables x, y, and z. It is denoted by:
(x-y-z)^2 = ?
Expansion of the Formula
To expand the (x-y-z)^2 formula, we need to follow the order of operations (PEMDAS) and apply the exponentiation rule. The expansion is as follows:
(x-y-z)^2 = (x-y-z)(x-y-z)
= x^2 - 2xy - 2xz + y^2 + 2yz - z^2
As you can see, the expansion involves the product of the binomial (x-y-z) with itself, resulting in a quadratic expression with six terms.
Examples and Applications
Let's consider a few examples to illustrate the application of the (x-y-z)^2 formula:
Example 1:
Find the value of (2-3-4)^2.
Using the formula, we get:
(2-3-4)^2 = 2^2 - 2(2)(3) - 2(2)(4) + 3^2 + 2(3)(4) - 4^2
= 4 - 12 - 16 + 9 + 24 - 16
= 25
Example 2:
Simplify the expression (x-2y+z)^2.
Using the formula, we get:
(x-2y+z)^2 = x^2 - 2x(2y) - 2x(z) + (2y)^2 + 2(2y)(z) - z^2
= x^2 - 4xy - 2xz + 4y^2 + 4yz - z^2
As you can see, the (x-y-z)^2 formula is a powerful tool for simplifying complex algebraic expressions and has numerous applications in mathematics, physics, and engineering.
Conclusion
In conclusion, the (x-y-z)^2 formula is a fundamental concept in algebra that has numerous applications in various fields. By understanding the expansion and application of this formula, you can simplify complex expressions and solve problems with ease.
