24. The curve $$ y=a x^{2}+b x+c $$ passes through the points $$ (1,4) $$, $$ (-2,19) $$ and $$ (0,5) $$. Find the equation of the curve.

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To determine the equation of the quadratic curve $$ y = ax^{2} + bx + c $$ that passes through the points $$ (1,4) $$, $$ (-2,19) $$, and $$ (0,5) $$, we can set up a system of equations based on these points.

1. **Using the point $$ (1,4) $$:**
   $$
   4 = a(1)^2 + b(1) + c \
   \Rightarrow a + b + c = 4 \quad \text{(Equation 1)}
   $$

2. **Using the point $$ (-2,19) $$:**
   $$
   19 = a(-2)^2 + b(-2) + c \
   \Rightarrow 4a - 2b + c = 19 \quad \text{(Equation 2)}
   $$

3. **Using the point $$ (0,5) $$:**
   $$
   5 = a(0)^2 + b(0) + c \
   \Rightarrow c = 5 \quad \text{(Equation 3)}
   $$

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

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

- From Equation 2:
  $$
  4a - 2b + 5 = 19 \
  \Rightarrow 4a - 2b = 14 \
  \Rightarrow 2a - b = 7 \quad \text{(Equation 5)}
  $$

**Solving Equations 4 and 5:**

- From Equation 4:
  $$
  a + b = -1 \
  \Rightarrow b = -1 - a
  $$

- Substitute $$ b = -1 - a $$ into Equation 5:
  $$
  2a - (-1 - a) = 7 \
  \Rightarrow 2a + 1 + a = 7 \
  \Rightarrow 3a + 1 = 7 \
  \Rightarrow 3a = 6 \
  \Rightarrow a = 2
  $$

- Substitute $$ a = 2 $$ back into $$ b = -1 - a $$:
  $$
  b = -1 - 2 = -3
  $$

**Final Equation:**

$$
y = 2x^{2} - 3x + 5
$$

**Answer:**  
After simplifying, an equation for the curve is y = 2 x² – 3 x + 5
The equation of the curve is $$ y = 2x^{2} - 3x + 5 $$.

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

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