https://r2.erweima.ai/mathbot/image/172658427579125dsevdr.jpg

Let's break down the task from the image you provided, focusing on determining the coefficients \(a\), \(b\), and \(c\) of the quadratic function.

We are given the quadratic equation \(y = ax^2 + bx + c\), and the parabola passes through three points:
- \(A(1, 4)\)
- \(B(-2, -5)\)
- \(C(3, 0)\)

This means that substituting the coordinates of these points into the equation \(y = ax^2 + bx + c\) will give us three equations that form a system of linear equations. Let's write these equations:

### Step 1: Deriving the system of equations
1. For \(A(1, 4)\): 
   \[
   a(1)^2 + b(1) + c = 4 \quad \Rightarrow \quad a + b + c = 4
   \]
2. For \(B(-2, -5)\): 
   \[
   a(-2)^2 + b(-2) + c = -5 \quad \Rightarrow \quad 4a - 2b + c = -5
   \]
3. For \(C(3, 0)\): 
   \[
   a(3)^2 + b(3) + c = 0 \quad \Rightarrow \quad 9a + 3b + c = 0
   \]

Thus, the system of equations becomes:
\[
\begin{aligned}
1. \quad &a + b + c = 4 \\
2. \quad &4a - 2b + c = -5 \\
3. \quad &9a + 3b + c = 0
\end{aligned}
\]

### Step 2: Matrix Form Representation
This system can be represented in matrix form \(A X = B\), where:

- Matrix \(A\) (from the system coefficients):
  \[
  A = \begin{pmatrix}
  1 & 1 & 1 \\
  4 & -2 & 1 \\
  9 & 3 & 1
  \end{pmatrix}
  \]
- Vector \(X\) (containing the unknowns \(a\), \(b\), and \(c\)):
  \[
  X = \begin{pmatrix}
  a \\
  b \\
  c
  \end{pmatrix}
  \]
- Vector \(B\) (the constants from the system):
  \[
  B = \begin{pmatrix}
  4 \\
  -5 \\
  0
  \end{pmatrix}
  \]

### Step 3: Solve the system
To find \(X\) (which contains \(a\), \(b\), and \(c\)), we can solve the equation \(X = A^{-1}B\), where \(A^{-1}\) is the inverse of the matrix \(A\).

Next, I will calculate the inverse of matrix \(A\) and multiply it by vector \(B\) to find the values of \(a\), \(b\), and \(c\). Let's compute that now.The solution to the system is:

\[
a = -1, \quad b = 2, \quad c = 3
\]

Thus, the equation of the parabola is:

\[
y = -x^2 + 2x + 3
\]

Would you like further details on any step, or do you have any questions?

Here are 5 related questions to deepen your understanding:
1. How do you calculate the inverse of a matrix manually?
2. What are the conditions for a matrix to be invertible?
3. How does substituting the points into the equation \(y = ax^2 + bx + c\) generate the system of equations?
4. Can a parabola be uniquely determined by fewer than three points?
5. How can you verify the solution by substituting the values of \(a\), \(b\), and \(c\) back into the system?

**Tip:** When solving a system of equations, always check if the matrix is invertible by ensuring its determinant is non-zero.

We consider a parabola of equation y = ax^2 + bx + c, where a, b, and c are real numbers, with a ≠ 0. The parabola passes through the points A(1, 4), B(-2, -5), and C(3, 0). Determine the coefficients a, b, and c using matrix algebra.

Learn how to determine the coefficients of a quadratic equation y = ax^2 + bx + c by solving a system of linear equations using matrix inversion. The parabola passes through the points A(1, 4), B(-2, -5), and C(3, 0). Includes step-by-step calculations of matrix A, vector B, and the solution.

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