23. The curve $$ y=a x^{2}+b x+c $$ passes through the points $$ (1,8) $$, $$ (0,5) $$ and $$ (3,20) $$. Find the values of $$ a, b $$ and $$ c $$ and hence the equation of the curve.

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To determine the values of $$ a $$, $$ b $$, and $$ c $$ for the quadratic curve $$ y = a x^{2} + b x + c $$ that passes through the points $$ (1,8) $$, $$ (0,5) $$, and $$ (3,20) $$, we can set up a system of equations based on these points.

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

2. **Using the point $$ (0,5) $$:**
   $$
   a(0)^2 + b(0) + c = 5 \
   \Rightarrow c = 5 \quad \text{(Equation 2)}
   $$

3. **Using the point $$ (3,20) $$:**
   $$
   a(3)^2 + b(3) + c = 20 \
   \Rightarrow 9a + 3b + c = 20 \quad \text{(Equation 3)}
   $$

**Substituting $$ c = 5 $$ from Equation 2 into Equations 1 and 3:**

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

- **From Equation 3:**
  $$
  9a + 3b + 5 = 20 \
  \Rightarrow 9a + 3b = 15 \
  \Rightarrow 3a + b = 5 \quad \text{(Equation 5)}
  $$

**Solving the system of Equations 4 and 5:**

- **Subtract Equation 4 from Equation 5:**
  $$
  (3a + b) - (a + b) = 5 - 3 \
  \Rightarrow 2a = 2 \
  \Rightarrow a = 1
  $$

- **Substitute $$ a = 1 $$ back into Equation 4:**
  $$
  1 + b = 3 \
  \Rightarrow b = 2
  $$

**Therefore, the values are:**
$$
a = 1, \quad b = 2, \quad c = 5
$$

**Hence, the equation of the curve is:**
$$
y = x^{2} + 2x + 5
$$

**Answer:**

After simplifying, a = 1, b = 2, c = 5. Thus, the curve is y = x² + 2 x + 5.
$$ a = 1 $$, $$ b = 2 $$, $$ c = 5 $$. The equation of the curve is $$ y = x^{2} + 2x + 5 $$.

The curve passes through the points , and . Find the values of and and hence the equation of the curve.

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