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 $$ (0,5) $$:**
   $$
   5 = a(0)^2 + b(0) + c \implies c = 5
   $$

2. **Using the point $$ (1,8) $$:**
   $$
   8 = a(1)^2 + b(1) + 5 \implies a + b = 3
   $$

3. **Using the point $$ (3,20) $$:**
   $$
   20 = a(3)^2 + b(3) + 5 \implies 9a + 3b = 15 \implies 3a + b = 5
   $$

Now, we have the following system of equations:
$$
\begin{cases}
a + b = 3 \
3a + b = 5
\end{cases}
$$

Subtract the first equation from the second:
$$
(3a + b) - (a + b) = 5 - 3 \implies 2a = 2 \implies a = 1
$$

Substitute $$ a = 1 $$ into $$ a + b = 3 $$:
$$
1 + b = 3 \implies b = 2
$$

**Final Values:**
$$
a = 1, \quad b = 2, \quad c = 5
$$

**Equation of the Curve:**
$$
y = x^{2} + 2x + 5
$$

**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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