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We are given that \( P(x) \) is a second-degree polynomial with the following conditions:

- \( P(2) = 3 \)
- \( P'(2) = 8 \)
- \( P''(2) = 1 \)

We need to find the value of \( P(1) \).

### Step 1: General form of a second-degree polynomial
Since \( P(x) \) is a second-degree polynomial, we can express it as:
\[
P(x) = ax^2 + bx + c
\]
where \( a \), \( b \), and \( c \) are constants.

### Step 2: First and second derivatives of \( P(x) \)
- First derivative:
\[
P'(x) = 2ax + b
\]
- Second derivative:
\[
P''(x) = 2a
\]

### Step 3: Use the given conditions

#### Condition 1: \( P(2) = 3 \)
Substitute \( x = 2 \) into the polynomial:
\[
a(2^2) + b(2) + c = 3 \quad \Rightarrow \quad 4a + 2b + c = 3 \quad \text{(Equation 1)}
\]

#### Condition 2: \( P'(2) = 8 \)
Substitute \( x = 2 \) into the first derivative:
\[
2a(2) + b = 8 \quad \Rightarrow \quad 4a + b = 8 \quad \text{(Equation 2)}
\]

#### Condition 3: \( P''(2) = 1 \)
Since \( P''(x) = 2a \), we have:
\[
2a = 1 \quad \Rightarrow \quad a = \frac{1}{2}
\]

### Step 4: Solve the system of equations

#### From Equation 2:
\[
4\left(\frac{1}{2}\right) + b = 8 \quad \Rightarrow \quad 2 + b = 8 \quad \Rightarrow \quad b = 6
\]

#### From Equation 1:
\[
4\left(\frac{1}{2}\right) + 2(6) + c = 3 \quad \Rightarrow \quad 2 + 12 + c = 3 \quad \Rightarrow \quad c = 3 - 14 = -11
\]

### Step 5: Find \( P(1) \)
Now that we have \( a = \frac{1}{2} \), \( b = 6 \), and \( c = -11 \), the polynomial is:
\[
P(x) = \frac{1}{2}x^2 + 6x - 11
\]
Substitute \( x = 1 \):
\[
P(1) = \frac{1}{2}(1^2) + 6(1) - 11 = \frac{1}{2} + 6 - 11 = -\frac{9}{2}
\]

### Final Answer:
\[
P(1) = -\frac{9}{2}
\]

Would you like more details or further clarifications?

Here are some related questions:
1. How do higher-degree polynomials differ in their solutions?
2. What would the solution process look like for a third-degree polynomial?
3. Can we generalize the approach for polynomials of any degree?
4. What is the geometric interpretation of the derivatives used in this problem?
5. How do we solve a system of linear equations using matrices?

**Tip:** When solving for unknown coefficients in polynomials, always start with the simplest conditions like \( P(x) \) or its derivatives at specific points.

If P is a second-degree polynomial such that P(2) = 3, P'(2) = 8, and P''(2) = 1, find the value of P(1).

Learn how to find the value of P(1) for a second-degree polynomial using conditions on the function and its derivatives. This problem includes solving a system of equations and using basic differentiation.

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