https://file.aiquickdraw.com/mathbot/file/1726150792358n66fgjen.jpg

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.

Solve the system of equations generated from a quadratic function f(x) = ax^2 + bx + c with given conditions: f(1) = 6, f(2) = 12, and f(3) = 22.

Learn how to solve a system of linear equations generated by the quadratic function f(x) = ax^2 + bx + c, with given values f(1) = 6, f(2) = 12, and f(3) = 22. This problem involves finding the coefficients a, b, and c using algebraic methods.

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