(x+y+z)^2+(x+y/2+z/3)^2-(x/2+y/3+z/4)^2

3 min read Jun 17, 2024

Expanding and Simplifying the Expression (x+y+z)^2+(x+y/2+z/3)^2-(x/2+y/3+z/4)^2

This expression involves squaring trinomials and subtracting them. To simplify it, we'll expand each squared term and then combine like terms.

Step 1: Expand each squared term

(x+y+z)^2: Using the formula (a+b+c)^2 = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc, we get: x^2 + y^2 + z^2 + 2xy + 2xz + 2yz
(x+y/2+z/3)^2: Applying the same formula: x^2 + (y/2)^2 + (z/3)^2 + 2x(y/2) + 2x(z/3) + 2(y/2)(z/3) = x^2 + y^2/4 + z^2/9 + xy + 2xz/3 + yz/3
(x/2+y/3+z/4)^2: Again, using the formula: (x/2)^2 + (y/3)^2 + (z/4)^2 + 2(x/2)(y/3) + 2(x/2)(z/4) + 2(y/3)(z/4) = x^2/4 + y^2/9 + z^2/16 + xy/3 + xz/4 + yz/6

Step 2: Combine the expanded terms

Now, let's combine the expanded terms, keeping track of the coefficients:

(x^2 + y^2 + z^2 + 2xy + 2xz + 2yz)

Step 3: Simplify by combining like terms

x^2 terms: x^2 + x^2 - x^2/4 = (15/4)x^2
y^2 terms: y^2 + y^2/4 - y^2/9 = (35/36)y^2
z^2 terms: z^2 + z^2/9 - z^2/16 = (143/144)z^2
xy terms: 2xy + xy - xy/3 = (8/3)xy
xz terms: 2xz + 2xz/3 - xz/4 = (25/12)xz
yz terms: 2yz + yz/3 - yz/6 = (13/6)yz

Final Result

The simplified expression is:

(15/4)x^2 + (35/36)y^2 + (143/144)z^2 + (8/3)xy + (25/12)xz + (13/6)yz