[tex]f(x) = -6x^2 + 7x + 4[/tex]
Step-by-step explanation:
For x = 1
[tex]a(1)^2 + b(1) + c = 5 \Rightarrow a + b + c = 5[/tex] (1)
For x = -3,
[tex]a(-3)^2 + b(-3) + c = -71[/tex]
[tex]\Rightarrow 9a - 3b + c = -71[/tex] (2)
For x = 3,
[tex]a(3)^2 + b(3) + c = -29[/tex]
[tex]\Rightarrow 9a + 3b + c = -29[/tex] (3)
Let's add Eqn(2) and Eqn(3) together to eliminate b:
[tex]\Rightarrow 18a + 2c = -100[/tex] (4)
Next, multiply Eqn(1) by 3 and then add to Eqn(2) to get
[tex]\Rightarrow 12a + 4c = -56[/tex] (5)
Then we want to eliminate c by multiplying Eqn(4) by -2 and then adding to Eqn(5) and we will get
[tex]-24a = 144 \Rightarrow a = -6[/tex]
Now that we have the value for a, we can use either Eqn(4) or Eqn(5) to solve for c and we will get
[tex]12(-6) + 4c = -56 \Rightarrow c = 4[/tex]
Using the values of a and c on Eqn(1), we find that [tex]b = 7[/tex] so therefore, our function f(x) has a form
[tex]f(x) = -6x^2 + 7x + 4[/tex]
