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To solve for the equation of the parabola in the form \( y = ax^2 + bx + c \) that passes through the points \( (-3, 43), (5, 27), (-4, 63) \), we need to find the coefficients \( a \), \( b \), and \( c \).

We will substitute each point into the general equation and solve the resulting system of three equations.

### Step 1: Set up the system of equations

1. For the point \( (-3, 43) \):
   \[
   43 = a(-3)^2 + b(-3) + c \implies 43 = 9a - 3b + c
   \]

2. For the point \( (5, 27) \):
   \[
   27 = a(5)^2 + b(5) + c \implies 27 = 25a + 5b + c
   \]

3. For the point \( (-4, 63) \):
   \[
   63 = a(-4)^2 + b(-4) + c \implies 63 = 16a - 4b + c
   \]

### Step 2: Solve the system of equations

Now we have the following system of linear equations:
\[
\begin{aligned}
1. & \quad 9a - 3b + c = 43 \\
2. & \quad 25a + 5b + c = 27 \\
3. & \quad 16a - 4b + c = 63 \\
\end{aligned}
\]

Let me solve this system to find \( a \), \( b \), and \( c \).The solution to the system of equations is:

\[
a = 2, \quad b = -6, \quad c = 7
\]

Thus, the equation of the parabola that passes through the points \((-3, 43)\), \((5, 27)\), and \((-4, 63)\) is:

\[
y = 2x^2 - 6x + 7
\]

Would you like more details on how this was solved or any further questions?

### Related Questions:
1. How can we verify that this parabola passes through the given points?
2. What is the vertex of the parabola \(y = 2x^2 - 6x + 7\)?
3.

Find an equation in the form y = ax^2 + bx + c for the parabola passing through the points (-3, 43), (5, 27), (-4, 63).

Learn how to find the equation of a parabola in the form y = ax^2 + bx + c by solving a system of equations using three given points: (-3, 43), (5, 27), and (-4, 63). This step-by-step solution covers key algebraic concepts and techniques, ideal for students in Grades 9-11.

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