To convert the polar equation $r = 2\cos\theta + 2\sin\theta$ to rectangular form, we use the following relationships between polar and rectangular coordinates:

1. $x = r\cos\theta$
2. $y = r\sin\theta$
3. $r = \sqrt{x^2 + y^2}$
4. $\cos\theta = \frac{x}{r}$, $\sin\theta = \frac{y}{r}$

Steps to Convert:

The given equation is: $r = 2\cos\theta + 2\sin\theta$

1. Multiply through by $r$ to eliminate the fraction involving $\cos\theta$ and $\sin\theta$: $r^2 = 2r\cos\theta + 2r\sin\theta$

2. Substitute $r^2 = x^2 + y^2$, $r\cos\theta = x$, and $r\sin\theta = y$: $x^2 + y^2 = 2x + 2y$

3. Rearrange the equation: $x^2 + y^2 - 2x - 2y = 0$

4. Complete the square for $x$ and $y$:

   • For $x^2 - 2x$, add and subtract $\left(\frac{2}{2}\right)^2 = 1$.
   • For $y^2 - 2y$, add and subtract $\left(\frac{2}{2}\right)^2 = 1$.

$(x^2 - 2x + 1) + (y^2 - 2y + 1) = 0 + 1 + 1$

5. Simplify to form perfect squares: $(x - 1)^2 + (y - 1)^2 = 2$

Final Answer:

The rectangular form of the equation is: $(x - 1)^2 + (y - 1)^2 = 2$

This represents a circle with center $(1, 1)$ and radius $\sqrt{2}$.

---

Would you like me to explain the completion of the square in more detail?

Related Questions:

1. What is the general form of a circle's equation in rectangular coordinates?
2. How can you convert a polar equation involving $r^2$ directly to rectangular form?
3. Why do we use the substitution $r^2 = x^2 + y^2$ during this process?
4. How would the process change if the coefficients of $\cos\theta$ and $\sin\theta$ were different?
5. How can you determine the center and radius of a circle from its equation?

Tip: Completing the square is a fundamental technique in algebra, especially for converting conic sections into standard forms. Practice often to master it!