If `p(x)=ax^(2)+bx` and `q(x)=lx^(2)+mx+n` with `p(1)=q(1), p(2)-q(2)=1`, and `p(3)-q(3)=4`, then `p(4)-q(4)` is equal to

Options

A. 7
B. 16
C. 9
D. none of these

Answer

To solve the problem step by step, we have the functions \( p(x) = ax^2 + bx \) and \( q(x) = lx^2 + mx + n \). We are given three conditions:

1. \( p(1) = q(1) \)
2. \( p(2) - q(2) = 1 \)
3. \( p(3) - q(3) = 4 \)

We need to find \( p(4) - q(4) \).

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

**Condition 1: \( p(1) = q(1) \)**

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

**Condition 2: \( p(2) - q(2) = 1 \)**

Calculating \( p(2) \) and \( q(2) \):
\[ p(2) = a(2)^2 + b(2) = 4a + 2b \]
\[ q(2) = l(2)^2 + m(2) + n = 4l + 2m + n \]
Setting up the equation gives us:
\[ (4a + 2b) - (4l + 2m + n) = 1 \implies 4a + 2b - 4l - 2m - n = 1 \quad \text{(Equation 2)} \]

**Condition 3: \( p(3) - q(3) = 4 \)**

Calculating \( p(3) \) and \( q(3) \):
\[ p(3) = a(3)^2 + b(3) = 9a + 3b \]
\[ q(3) = l(3)^2 + m(3) + n = 9l + 3m + n \]
Setting up the equation gives us:
\[ (9a + 3b) - (9l + 3m + n) = 4 \implies 9a + 3b - 9l - 3m - n = 4 \quad \text{(Equation 3)} \]

### Step 2: Solve the system of equations

We have three equations:
1. \( a + b = l + m + n \)
2. \( 4a + 2b - 4l - 2m - n = 1 \)
3. \( 9a + 3b - 9l - 3m - n = 4 \)

**Rearranging Equation 1:**
From Equation 1, we can express \( n \):
\[ n = a + b - l - m \quad \text{(Substituting this in Equations 2 and 3)} \]

**Substituting \( n \) in Equation 2:**
\[ 4a + 2b - 4l - 2m - (a + b - l - m) = 1 \]
Simplifying this:
\[ 4a + 2b - 4l - 2m - a - b + l + m = 1 \]
\[ (4a - a) + (2b - b) + (-4l + l) + (-2m + m) = 1 \]
\[ 3a + b - 3l - m = 1 \quad \text{(Equation 4)} \]

**Substituting \( n \) in Equation 3:**
\[ 9a + 3b - 9l - 3m - (a + b - l - m) = 4 \]
Simplifying this:
\[ 9a + 3b - 9l - 3m - a - b + l + m = 4 \]
\[ (9a - a) + (3b - b) + (-9l + l) + (-3m + m) = 4 \]
\[ 8a + 2b - 8l - 2m = 4 \quad \text{(Equation 5)} \]

### Step 3: Solve Equations 4 and 5

From Equation 4:
\[ 3a + b - 3l - m = 1 \]
From Equation 5:
\[ 8a + 2b - 8l - 2m = 4 \]

**Multiply Equation 4 by 2:**
\[ 6a + 2b - 6l - 2m = 2 \quad \text{(Equation 6)} \]

**Now subtract Equation 6 from Equation 5:**
\[ (8a + 2b - 8l - 2m) - (6a + 2b - 6l - 2m) = 4 - 2 \]
This simplifies to:
\[ 2a - 2l = 2 \implies a - l = 1 \quad \text{(Equation 7)} \]

### Step 4: Substitute back to find values

Substituting \( l = a - 1 \) back into Equation 4:
\[ 3a + b - 3(a - 1) - m = 1 \]
This simplifies to:
\[ 3a + b - 3a + 3 - m = 1 \implies b - m + 3 = 1 \implies b - m = -2 \quad \text{(Equation 8)} \]

Now we have two equations:
1. \( a - l = 1 \)
2. \( b - m = -2 \)

### Step 5: Calculate \( p(4) - q(4) \)

To find \( p(4) - q(4) \):
\[ p(4) = a(4^2) + b(4) = 16a + 4b \]
\[ q(4) = l(4^2) + m(4) + n = 16l + 4m + n \]
Thus,
\[ p(4) - q(4) = (16a + 4b) - (16l + 4m + n) \]

Substituting \( l = a - 1 \) and \( n = a + b - l - m \):
\[ p(4) - q(4) = 16a + 4b - [16(a - 1) + 4m + (a + b - (a - 1) - m)] \]
Simplifying this expression will yield the final result.

### Final Calculation

After performing the calculations, we find:
\[ p(4) - q(4) = 9 \]

### Conclusion
Thus, the final answer is:
\[ \boxed{9} \]