25. The curve $$ y=a x^{2}+b x+c $$ passes through $$ (1,8),(-1,2) $$ and $$ (2,14) $$. Find the equation of the curve.

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To find the equation of the curve $$ y = a x^{2} + b x + c $$ that passes through the points $$ (1,8) $$, $$ (-1,2) $$, and $$ (2,14) $$, we'll set up a system of equations based on these points.

1. **Using the point (1, 8):**
   $$
   a(1)^2 + b(1) + c = 8 \quad \Rightarrow \quad a + b + c = 8 \quad \text{(Equation 1)}
   $$

2. **Using the point (-1, 2):**
   $$
   a(-1)^2 + b(-1) + c = 2 \quad \Rightarrow \quad a - b + c = 2 \quad \text{(Equation 2)}
   $$

3. **Using the point (2, 14):**
   $$
   a(2)^2 + b(2) + c = 14 \quad \Rightarrow \quad 4a + 2b + c = 14 \quad \text{(Equation 3)}
   $$

**Solving the System of Equations:**

- **Subtract Equation 2 from Equation 1:**
  $$
  (a + b + c) - (a - b + c) = 8 - 2 \quad \Rightarrow \quad 2b = 6 \quad \Rightarrow \quad b = 3
  $$

- **Substitute $$ b = 3 $$ into Equation 1:**
  $$
  a + 3 + c = 8 \quad \Rightarrow \quad a + c = 5 \quad \text{(Equation 4)}
  $$

- **Substitute $$ b = 3 $$ into Equation 3:**
  $$
  4a + 6 + c = 14 \quad \Rightarrow \quad 4a + c = 8 \quad \text{(Equation 5)}
  $$

- **Subtract Equation 4 from Equation 5:**
  $$
  (4a + c) - (a + c) = 8 - 5 \quad \Rightarrow \quad 3a = 3 \quad \Rightarrow \quad a = 1
  $$

- **Substitute $$ a = 1 $$ into Equation 4:**
  $$
  1 + c = 5 \quad \Rightarrow \quad c = 4
  $$

**Final Equation:**
$$
y = x^{2} + 3x + 4
$$

**Answer:**  
After simplifying, an equation for the curve is y = x² + 3 x + 4
The equation of the curve is $$ y = x^{2} + 3x + 4 $$.

25. The curve \( y=a x^{2}+b x+c \) passes through \( (1,8),(-1,2) \) and \( (2,14) \). Find the equation of the curve.

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