Using matrix method, find the quadratic defined by `f(x) = ax^(2) + bx + c` if f(1) = 0, f(2) = -2 and f(3) = -6. Hence find the value of f(-1).

Answer

To solve the problem using the matrix method, we need to find the coefficients \( a \), \( b \), and \( c \) of the quadratic function \( f(x) = ax^2 + bx + c \) given the conditions \( f(1) = 0 \), \( f(2) = -2 \), and \( f(3) = -6 \).

### Step 1: Set up the equations based on the given conditions

From the conditions provided, we can create the following equations:

1. \( f(1) = a(1)^2 + b(1) + c = 0 \)
This simplifies to:
\[ a + b + c = 0 \]
(Equation 1)

2. \( f(2) = a(2)^2 + b(2) + c = -2 \)
This simplifies to:
\[ 4a + 2b + c = -2 \]
(Equation 2)

3. \( f(3) = a(3)^2 + b(3) + c = -6 \)
This simplifies to:
\[ 9a + 3b + c = -6 \]
(Equation 3)

### Step 2: Write the equations in matrix form

We can express the system of equations in matrix form as follows:

\[ \begin{bmatrix} 1 & 1 & 1 \\ 4 & 2 & 1 \\ 9 & 3 & 1 \end{bmatrix} \begin{bmatrix} a \\ b \\ c \end{bmatrix} = \begin{bmatrix} 0 \\ -2 \\ -6 \end{bmatrix} \]

### Step 3: Solve the system of equations

We will now solve the system using elimination or substitution.

**Subtract Equation 1 from Equation 2:**

\[ (4a + 2b + c) - (a + b + c) = -2 - 0 \]
This simplifies to:
\[ 3a + b = -2 \quad \text{(Equation 4)} \]

**Subtract Equation 1 from Equation 3:**

\[ (9a + 3b + c) - (a + b + c) = -6 - 0 \]
This simplifies to:
\[ 8a + 2b = -6 \quad \text{(Equation 5)} \]

### Step 4: Simplify Equation 5

Divide Equation 5 by 2:
\[ 4a + b = -3 \quad \text{(Equation 6)} \]

### Step 5: Solve Equations 4 and 6

Now, we have two equations:
1. \( 3a + b = -2 \) (Equation 4)
2. \( 4a + b = -3 \) (Equation 6)

Subtract Equation 4 from Equation 6:
\[ (4a + b) - (3a + b) = -3 + 2 \]
This simplifies to:
\[ a = -1 \]

### Step 6: Substitute \( a \) back to find \( b \)

Substituting \( a = -1 \) into Equation 4:
\[ 3(-1) + b = -2 \]
This simplifies to:
\[ -3 + b = -2 \implies b = 1 \]

### Step 7: Substitute \( a \) and \( b \) back to find \( c \)

Substituting \( a = -1 \) and \( b = 1 \) into Equation 1:
\[ -1 + 1 + c = 0 \]
This simplifies to:
\[ c = 0 \]

### Step 8: Write the quadratic function

Now we have:
- \( a = -1 \)
- \( b = 1 \)
- \( c = 0 \)

Thus, the quadratic function is:
\[ f(x) = -x^2 + x \]

### Step 9: Find \( f(-1) \)

Now we need to find the value of \( f(-1) \):
\[ f(-1) = -(-1)^2 + (-1) = -1 - 1 = -2 \]

### Final Answer
The value of \( f(-1) \) is \( -2 \).

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