If `f(x)=ax^2+bx+c` and `g(x)=px^2+qx` with `g(1)=f(1)` , `g(2)-f(2)=1`and `g(3)-f(3)=4` then `g(4)-f(4)` is

Options

A. 0
B. 5
C. 6
D. none of these

Explanation

To solve the problem step by step, we will analyze the given functions and the conditions provided.

### Given:
1. \( f(x) = ax^2 + bx + c \)
2. \( g(x) = px^2 + qx \)
3. \( g(1) = f(1) \)
4. \( g(2) - f(2) = 1 \)
5. \( g(3) - f(3) = 4 \)

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

**Condition 1: \( g(1) = f(1) \)**

Calculating \( f(1) \) and \( g(1) \):
\[ f(1) = a(1)^2 + b(1) + c = a + b + c \]
\[ g(1) = p(1)^2 + q(1) = p + q \]
Setting them equal gives us:
\[ a + b + c = p + q \quad \text{(Equation 1)} \]

**Condition 2: \( g(2) - f(2) = 1 \)**

Calculating \( f(2) \) and \( g(2) \):
\[ f(2) = a(2)^2 + b(2) + c = 4a + 2b + c \]
\[ g(2) = p(2)^2 + q(2) = 4p + 2q \]
Setting up the equation:
\[ (4p + 2q) - (4a + 2b + c) = 1 \]
This simplifies to:
\[ 4p + 2q - 4a - 2b - c = 1 \quad \text{(Equation 2)} \]

**Condition 3: \( g(3) - f(3) = 4 \)**

Calculating \( f(3) \) and \( g(3) \):
\[ f(3) = a(3)^2 + b(3) + c = 9a + 3b + c \]
\[ g(3) = p(3)^2 + q(3) = 9p + 3q \]
Setting up the equation:
\[ (9p + 3q) - (9a + 3b + c) = 4 \]
This simplifies to:
\[ 9p + 3q - 9a - 3b - c = 4 \quad \text{(Equation 3)} \]

### Step 2: Rewrite the equations

From Equation 1:
\[ p + q = a + b + c \quad \text{(Rearranging gives us \( p - a + q - b - c = 0 \))} \]

Let:
\[ A = p - a, \quad B = q - b \]
Then:
\[ A + B - c = 0 \quad \text{(Equation 4)} \]

### Step 3: Substitute \( A \) and \( B \) into Equations 2 and 3

**From Equation 2:**
\[ 4A + 2B - c = 1 \quad \text{(Equation 5)} \]

**From Equation 3:**
\[ 9A + 3B - c = 4 \quad \text{(Equation 6)} \]

### Step 4: Solve the equations

**Subtract Equation 5 from Equation 6:**
\[ (9A + 3B - c) - (4A + 2B - c) = 4 - 1 \]
This simplifies to:
\[ 5A + B = 3 \quad \text{(Equation 7)} \]

Now, we can express \( B \) in terms of \( A \):
\[ B = 3 - 5A \quad \text{(Equation 8)} \]

**Substituting Equation 8 into Equation 5:**
\[ 4A + 2(3 - 5A) - c = 1 \]
This simplifies to:
\[ 4A + 6 - 10A - c = 1 \]
\[ -6A - c + 6 = 1 \]
\[ -6A - c = -5 \quad \Rightarrow \quad c = -6A + 5 \quad \text{(Equation 9)} \]

### Step 5: Substitute \( A \) and \( B \) back into Equation 4

Using Equation 4:
\[ A + (3 - 5A) - (-6A + 5) = 0 \]
This simplifies to:
\[ A + 3 - 5A + 6A - 5 = 0 \]
\[ 2A - 2 = 0 \quad \Rightarrow \quad A = 1 \]

### Step 6: Find \( B \) and \( c \)

Substituting \( A = 1 \) into Equation 8:
\[ B = 3 - 5(1) = -2 \]

Substituting \( A = 1 \) into Equation 9:
\[ c = -6(1) + 5 = -1 \]

### Step 7: Find \( g(4) - f(4) \)

Now we can find \( g(4) - f(4) \):
\[ g(4) - f(4) = 16A + 4B - c \]
Substituting the values:
\[ = 16(1) + 4(-2) - (-1) \]
\[ = 16 - 8 + 1 = 9 \]

### Final Answer:
\[ g(4) - f(4) = 9 \]