Find the values of a,b and c by matrix method so that the graph of the polynomial p (x)=ax^2 + bx + c passes through the points (1,2),(-1,6) and (2,3).
To find the values of a, b, and c using the matrix method, we need to set up a system of equations based on the given points.
The equation of the polynomial p(x) = ax^2 + bx + c can be rewritten in matrix form as:
⎡ 1 1 1 ⎤ ⎡ a ⎤ ⎡ 2 ⎤
⎢ 1 -1 1 ⎥ * ⎢ b ⎥ = ⎢ 6 ⎥
⎣ 4 2 1 ⎦ ⎣ c ⎦ ⎣ 3 ⎦
To solve the system, we need to find the inverse of the coefficient matrix:
⎡ 1 1 1 ⎤^(-1)
To find the inverse, we need to calculate the determinant of the coefficient matrix:
det ⎡ 1 1 1 ⎤ = 1(1*1-1*1) - 1(1*1-1*1) + 1(1*-1-1*1) = 0
Since the determinant is 0, the matrix is singular, meaning it doesn't have an inverse. This implies that there is no unique solution for the system, and we cannot find the values of a, b, and c using the matrix method.
However, we can still find a solution by setting up and solving a system of equations using the given points.
1. Substitute the first point (1, 2) into the polynomial equation:
a(1)^2 + b(1) + c = 2
a + b + c = 2
2. Substitute the second point (-1, 6) into the polynomial equation:
a(-1)^2 + b(-1) + c = 6
a - b + c = 6
3. Substitute the third point (2, 3) into the polynomial equation:
a(2)^2 + b(2) + c = 3
4a + 2b + c = 3
Now, we have a system of three equations:
a + b + c = 2
a - b + c = 6
4a + 2b + c = 3
Solve this system of equations using any applicable method (such as substitution or elimination) to find the values of a, b, and c.