We are given the function:

$f(x) = ax^2 - 5x + c$

with the conditions:

• $f(1) = 0$
• $f(2) = 1$
• $f(3) = 6$

We need to determine the values of $a$ and $c$, and verify the result for $f(3)$.

Step 1: Substitute $f(1) = 0$ into the function

$f(1) = a(1)^2 - 5(1) + c = 0$ $a - 5 + c = 0 \implies a + c = 5 \quad \text{(Equation 1)}$

Step 2: Substitute $f(2) = 1$ into the function

$f(2) = a(2)^2 - 5(2) + c = 1$ $4a - 10 + c = 1 \implies 4a + c = 11 \quad \text{(Equation 2)}$

Step 3: Solve the system of equations

From Equation 1: $c = 5 - a$

Substitute this into Equation 2:

$4a + (5 - a) = 11$ $4a - a + 5 = 11 \implies 3a = 6 \implies a = 2$

Substitute $a = 2$ back into Equation 1:

$2 + c = 5 \implies c = 3$

Step 4: Verify $f(3) = 6$

Now, substitute $a = 2$ and $c = 3$ into the function:

$f(3) = 2(3)^2 - 5(3) + 3 = 2(9) - 15 + 3 = 18 - 15 + 3 = 6$

Thus, $f(3) = 6$, which confirms the solution.

Let me know if you'd like further details, or have any questions!

Here are some related questions:

1. How can you determine the general form of a quadratic equation?
2. What are the steps to solve a system of linear equations like in this problem?
3. How would the solution change if one condition was modified, for example, if $f(2) = 2$?
4. Can we solve this type of problem graphically, and how would the graph look?
5. How do changes in the coefficients of the quadratic function affect the graph's shape and position?

Tip: When solving systems of equations, always check that your final solution satisfies all given conditions.