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.

To find the values of \( a \), \( b \), and \( c \), we need to create a system of equations using the points given. 1. **Using the point (1, 8)**: \[ 8 = a(1)^2 + b(1) + c \implies a + b + c = 8 \quad (1) \] 2. **Using the point (0, 5)**: \[ 5 = a(0)^2 + b(0) + c \implies c = 5 \quad (2) \] 3. **Using the point (3, 20)**: \[ 20 = a(3)^2 + b(3) + c \implies 9a + 3b + c = 20 \quad (3) \] Now, substituting \( c = 5 \) from equation (2) into equations (1) and (3): Substituting into equation (1): \[ a + b + 5 = 8 \implies a + b = 3 \quad (4) \] Substituting into equation (3): \[ 9a + 3b + 5 = 20 \implies 9a + 3b = 15 \quad (5) \] Now we have a simpler system of equations to solve: From equation (4): \( b = 3 - a \) Substituting \( b = 3 - a \) into equation (5): \[ 9a + 3(3 - a) = 15 \\ 9a + 9 - 3a = 15 \\ 6a + 9 = 15 \\ 6a = 6 \\ a = 1 \] Now, substitute \( a = 1 \) back into equation (4) to find \( b \): \[ 1 + b = 3 \implies b = 2 \] We've now found \( a \), \( b \), and \( c \): \[ a = 1, \quad b = 2, \quad c = 5 \] Now we can write the equation of the curve: \[ y = x^2 + 2x + 5 \]