If `f(x)=ax^2+bx+2`, where `f(-1)=7 and f(1)=5,` then: `f(a,b)-=`

Options

A. `(4,-1)`
B. `(1,-4)`
C. `(4,1)`
D. none of these.

Answer

To solve the problem step by step, we need to find the values of \( a \) and \( b \) using the given conditions for the function \( f(x) = ax^2 + bx + 2 \).

### Step 1: Set up the equations using the given values of \( f(-1) \) and \( f(1) \).

1. We know that \( f(-1) = 7 \).
\[ f(-1) = a(-1)^2 + b(-1) + 2 = 7 \]
Simplifying this gives:
\[ a - b + 2 = 7 \]
Rearranging, we get:
\[ a - b = 5 \quad \text{(Equation 1)} \]

2. Next, we know that \( f(1) = 5 \).
\[ f(1) = a(1)^2 + b(1) + 2 = 5 \]
Simplifying this gives:
\[ a + b + 2 = 5 \]
Rearranging, we get:
\[ a + b = 3 \quad \text{(Equation 2)} \]

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

Now we have two equations:
1. \( a - b = 5 \) (Equation 1)
2. \( a + b = 3 \) (Equation 2)

We can add these two equations to eliminate \( b \):
\[ (a - b) + (a + b) = 5 + 3 \]
This simplifies to:
\[ 2a = 8 \]
So, dividing both sides by 2:
\[ a = 4 \]

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

Now that we have \( a = 4 \), we can substitute this value into Equation 2:
\[ 4 + b = 3 \]
Rearranging gives:
\[ b = 3 - 4 = -1 \]

### Step 4: Find \( f(a, b) \).

Now we have \( a = 4 \) and \( b = -1 \). We need to find \( f(a, b) \):
\[ f(x) = 4x^2 - x + 2 \]
To find \( f(a, b) \), we substitute \( a \) and \( b \) into the function:
\[ f(4, -1) = 4(4)^2 + (-1)(4) + 2 \]
Calculating this gives:
\[ = 4(16) - 4 + 2 \]
\[ = 64 - 4 + 2 \]
\[ = 62 \]

### Final Answer:
Thus, \( f(a, b) = 62 \).

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