%5E2-formula.jpeg "(x+y+z)^2 Formula")
The Formula for (x+y+z)^2: A Comprehensive Guide
In algebra, the expansion of the square of a sum of three terms, namely (x+y+z)^2, is a fundamental concept that is widely used in various branches of mathematics, physics, and engineering. In this article, we will derive and explain the formula for (x+y+z)^2, along with its proof and examples.
The Formula:
The formula for (x+y+z)^2 is given by:
(x+y+z)^2 = x^2 + y^2 + z^2 + 2(xy + yz + zx)
This formula is a direct extension of the binomial theorem, which states that (a+b)^2 = a^2 + 2ab + b^2.
Proof:
To prove the formula, we can start with the definition of the square of a sum:
(x+y+z)^2 = (x+y+z)(x+y+z)
Expanding the right-hand side using the distributive property, we get:
(x+y+z)(x+y+z) = x(x+y+z) + y(x+y+z) + z(x+y+z)
= x^2 + xy + xz + yx + y^2 + yz + zx + zy + z^2
Combine like terms and simplify:
= x^2 + y^2 + z^2 + 2(xy + yz + zx)
Thus, we have derived the formula for (x+y+z)^2.
Examples:
Example 1:
Find the value of (2+3+4)^2 using the formula.
(2+3+4)^2 = 2^2 + 3^2 + 4^2 + 2(2*3 + 3*4 + 4*2) = 4 + 9 + 16 + 2(6 + 12 + 8) = 4 + 9 + 16 + 2(26) = 4 + 9 + 16 + 52 = 81
Example 2:
Find the value of (x+2y+z)^2 using the formula.
(x+2y+z)^2 = x^2 + (2y)^2 + z^2 + 2(x(2y) + (2y)z + zx) = x^2 + 4y^2 + z^2 + 2(2xy + 2yz + zx)
Example 3:
Find the value of (a+b+c)^2 when a = 1, b = 2, and c = 3.
(a+b+c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca) = 1^2 + 2^2 + 3^2 + 2(1*2 + 2*3 + 3*1) = 1 + 4 + 9 + 2(2 + 6 + 3) = 1 + 4 + 9 + 2(11) = 36
Conclusion:
In conclusion, the formula for (x+y+z)^2 is a powerful tool for expanding the square of a sum of three terms. It has numerous applications in various fields, including algebra, geometry, trigonometry, and physics. By understanding and applying this formula, we can solve a wide range of problems involving quadratic expressions and equations.
