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.

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