Find the equation of the parabola y = ax2 + bx + c through (1, 2), (2, 6), and (−1, 18). (I.e. find a, b and c.)

We are given three points \((1, 2)\), \((2, 6)\), and \((-1, 18)\) and need to find the coefficients \(a\), \(b\), and \(c\) in the general form of the parabola:

\[
y = ax^2 + bx + c
\]

Substitute each point into the equation to form a system of three equations.

1. For \((1, 2)\):

\[
2 = a(1)^2 + b(1) + c \quad \Rightarrow \quad a + b + c = 2
\]

2. For \((2, 6)\):

\[
6 = a(2)^2 + b(2) + c \quad \Rightarrow \quad 4a + 2b + c = 6
\]

3. For \((-1, 18)\):

\[
18 = a(-1)^2 + b(-1) + c \quad \Rightarrow \quad a - b + c = 18
\]

We now have the following system of equations:

\[
\begin{align*}
a + b + c &= 2 \quad \text{(1)} \\
4a + 2b + c &= 6 \quad \text{(2)} \\
a - b + c &= 18 \quad \text{(3)}
\end{align*}
\]

Let's solve this system step by step.

### Step 1: Subtract Equation (1) from Equation (2)
\[
(4a + 2b + c) - (a + b + c) = 6 - 2
\]
\[
3a + b = 4 \quad \text{(4)}
\]

### Step 2: Subtract Equation (1) from Equation (3)
\[
(a - b + c) - (a + b + c) = 18 - 2
\]
\[
-2b = 16 \quad \Rightarrow \quad b = -8 \quad \text{(5)}
\]

### Step 3: Substitute \(b = -8\) into Equation (4)
\[
3a + (-8) = 4
\]
\[
3a = 12 \quad \Rightarrow \quad a = 4 \quad \text{(6)}
\]

### Step 4: Substitute \(a = 4\) and \(b = -8\) into Equation (1)
\[
4 + (-8) + c = 2
\]
\[
-4 + c = 2 \quad \Rightarrow \quad c = 6 \quad \text{(7)}
\]

### Final Answer
The values of \(a\), \(b\), and \(c\) are:

\[
a = 4, \quad b = -8, \quad c = 6
\]

Thus, the equation of the parabola is:

\[
y = 4x^2 - 8x + 6
\]

Would you like more details or further explanation? 

Here are 5 related questions you might find useful:
1. How can you solve a system of equations using matrices?
2. What is the significance of the discriminant in quadratic equations?
3. How can you find the vertex of a parabola from its equation?
4. How does changing the coefficient \(a\) affect the shape of a parabola?
5. What are the roots of this parabola, and how can we find them?

**Tip:** When solving systems of equations, always check for consistent and independent solutions before proceeding to simplify!

Find the equation of the parabola y = ax^2 + bx + c through (1, 2), (2, 6), and (−1, 18). (I.e. find a, b and c.)

Learn how to find the equation of a parabola y = ax^2 + bx + c that passes through the points (1, 2), (2, 6), and (-1, 18). This problem involves solving a system of equations to determine the values of a, b, and c. Suitable for students in Grades 9-11.

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