If `f (x) =ax ^(2) + bx + 2 and f (1) =3, f (4) =42,` then a and b respectively are

Options

A. `-3,2`
B. `3,2`
C. `-2,3`
D. `3,-2`

Answer

To solve the problem, we need to find the values of \( a \) and \( b \) in the function \( f(x) = ax^2 + bx + 2 \) given the conditions \( f(1) = 3 \) and \( f(4) = 42 \).

### Step 1: Set up the equations using the given conditions.

1. **Using \( f(1) = 3 \)**:
\[ f(1) = a(1)^2 + b(1) + 2 = 3 \]
This simplifies to:
\[ a + b + 2 = 3 \]
Subtracting 2 from both sides:
\[ a + b = 1 \quad \text{(Equation 1)} \]

2. **Using \( f(4) = 42 \)**:
\[ f(4) = a(4)^2 + b(4) + 2 = 42 \]
This simplifies to:
\[ 16a + 4b + 2 = 42 \]
Subtracting 2 from both sides:
\[ 16a + 4b = 40 \]
Dividing the entire equation by 4:
\[ 4a + b = 10 \quad \text{(Equation 2)} \]

### Step 2: Solve the system of equations.

Now we have two equations:
1. \( a + b = 1 \) (Equation 1)
2. \( 4a + b = 10 \) (Equation 2)

We can solve these equations simultaneously.

**Subtract Equation 1 from Equation 2**:
\[ (4a + b) - (a + b) = 10 - 1 \]
This simplifies to:
\[ 3a = 9 \]
Dividing both sides by 3:
\[ a = 3 \]

### Step 3: Substitute the value of \( a \) back to find \( b \).

Now that we have \( a = 3 \), we can substitute this value back into Equation 1:
\[ 3 + b = 1 \]
Subtracting 3 from both sides gives:
\[ b = 1 - 3 = -2 \]

### Final Answer:
Thus, the values of \( a \) and \( b \) are:
\[ a = 3, \quad b = -2 \]