To solve the equation [tex]\((x-2)^2 + (y+2)^2\)[/tex], we can follow these steps:
1. Understand that [tex]\((x-2)^2\)[/tex] represents a term where [tex]\(x\)[/tex] is offset by 2, squared. Similarly, [tex]\((y+2)^2\)[/tex] represents a term where [tex]\(y\)[/tex] is offset by -2, squared.
2. The equation [tex]\((x-2)^2 + (y+2)^2\)[/tex] is already expanded.
Let's break it down further:
- [tex]\((x-2)^2\)[/tex]:
- This expression is a perfect square trinomial: [tex]\((x - 2) \cdot (x - 2)\)[/tex]
- Expanding this, we get: [tex]\(x^2 - 4x + 4\)[/tex]
- [tex]\((y+2)^2\)[/tex]:
- This expression is also a perfect square trinomial: [tex]\((y + 2) \cdot (y + 2)\)[/tex]
- Expanding this, we get: [tex]\(y^2 + 4y + 4\)[/tex]
So our combined expression is:
[tex]\[ x^2 - 4x + 4 + y^2 + 4y + 4 \][/tex]
However, note that these expansions are additional steps to better understand the components of the equation. The initial form provided itself,
[tex]\[ (x - 2)^2 + (y + 2)^2 \][/tex]
is already a simplified and correct representation of the sum of squares of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] terms offset by 2 and -2 respectively.
