To find the equation of the circle passing through the points $(3, 4)$, $(-5, -2)$, and $(3, -2)$, we start with the general form of the equation of a circle:

$x^2 + y^2 + ax + by = c$

Step 1: Substituting the points into the equation

For each point, substitute $(x, y)$ into the equation to form a system of equations.

1. Using $(3, 4)$: $3^2 + 4^2 + 3a + 4b = c$ $9 + 16 + 3a + 4b = c \quad \Rightarrow \quad 3a + 4b - c = -25 \quad \text{(Equation 1)}$

2. Using $(-5, -2)$: $(-5)^2 + (-2)^2 - 5a - 2b = c$ $25 + 4 - 5a - 2b = c \quad \Rightarrow \quad -5a - 2b - c = -29 \quad \text{(Equation 2)}$

3. Using $(3, -2)$: $3^2 + (-2)^2 + 3a - 2b = c$ $9 + 4 + 3a - 2b = c \quad \Rightarrow \quad 3a - 2b - c = -13 \quad \text{(Equation 3)}$

Step 2: Solve the system of equations

We now solve for $a$, $b$, and $c$ using these equations:

1. $3a + 4b - c = -25$
2. $-5a - 2b - c = -29$
3. $3a - 2b - c = -13$

Step 2.1: Eliminate $c$

Subtract Equation 3 from Equation 1: $(3a + 4b - c) - (3a - 2b - c) = -25 - (-13)$ $6b = -12 \quad \Rightarrow \quad b = -2$

Substitute $b = -2$ into Equation 1 and Equation 2:

1. From Equation 1: $3a + 4(-2) - c = -25$ $3a - 8 - c = -25 \quad \Rightarrow \quad 3a - c = -17 \quad \text{(Equation 4)}$

2. From Equation 2: $-5a - 2(-2) - c = -29$ $-5a + 4 - c = -29 \quad \Rightarrow \quad -5a - c = -33 \quad \text{(Equation 5)}$

Step 2.2: Solve for $a$ and $c$

Subtract Equation 4 from Equation 5: $(-5a - c) - (3a - c) = -33 - (-17)$ $-8a = -16 \quad \Rightarrow \quad a = 2$

Substitute $a = 2$ into Equation 4: $3(2) - c = -17$ $6 - c = -17 \quad \Rightarrow \quad c = 23$

Step 3: Write the equation of the circle

Now that we have $a = 2$, $b = -2$, and $c = 23$, the equation of the circle is:

$x^2 + y^2 + 2x - 2y = 23$

Would you like further explanation or to verify these results step-by-step? Here are five questions to deepen your understanding:

1. Why do we use the general equation of a circle to solve such problems?
2. How can the substitution method help solve systems of equations effectively?
3. What is the significance of each variable ($a, b, c$) in the circle equation?
4. Could you apply this method to find the equation of a circle with different points?
5. How would this process change if we used a matrix to solve the system of equations?

Tip: To confirm your circle equation, you can substitute the given points back into the equation and check if they satisfy it.