The image you provided contains a system of equations based on a quadratic function of the form:

$f(x) = ax^2 + bx + c$

The equations given are:

1. $f(1) = a(1)^2 + b(1) + c = 6$, which simplifies to: $a + b + c = 6 \quad \text{(i)}$

2. $f(2) = a(2)^2 + b(2) + c = 12$, which simplifies to: $4a + 2b + c = 12 \quad \text{(ii)}$

3. $f(3) = a(3)^2 + b(3) + c = 22$, which simplifies to: $9a + 3b + c = 22 \quad \text{(iii)}$

This gives us the following system of linear equations:

1. $a + b + c = 6$
2. $4a + 2b + c = 12$
3. $9a + 3b + c = 22$

Would you like me to solve this system of equations to find $a$, $b$, and $c$?

Let me know if you'd like any clarifications! Here's a set of related questions to help expand the understanding:

1. What methods can be used to solve systems of linear equations?
2. How is the discriminant of a quadratic equation calculated, and what does it represent?
3. Can quadratic functions have real or complex roots, and how are they determined?
4. How does the graph of a quadratic function change as the coefficients $a$, $b$, and $c$ vary?
5. How would you interpret the vertex and axis of symmetry of a quadratic function?

Tip: Solving a system of equations using substitution is effective when one variable is easily isolated.