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try to find the equation of the quadratic which passes through the points (-1,7) (2,1)and (3,-1). explain your answer

To find the equation of the quadratic that passes through the points \((-1, 7)\), \((2, 1)\), and \((3, -1)\), we can assume the general form of a quadratic equation:

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

We need to find the values of \(a\), \(b\), and \(c\) by substituting each of the given points into the equation. This will result in a system of three equations.

### Step 1: Write the equations for the three points

1. For \((-1, 7)\):

\[
7 = a(-1)^2 + b(-1) + c
\]
\[
7 = a - b + c \quad \text{(Equation 1)}
\]

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

\[
1 = a(2)^2 + b(2) + c
\]
\[
1 = 4a + 2b + c \quad \text{(Equation 2)}
\]

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

\[
-1 = a(3)^2 + b(3) + c
\]
\[
-1 = 9a + 3b + c \quad \text{(Equation 3)}
\]

### Step 2: Solve the system of equations

Now we solve these three simultaneous equations:

1. \(a - b + c = 7\)
2. \(4a + 2b + c = 1\)
3. \(9a + 3b + c = -1\)

Let me calculate the values of \(a\), \(b\), and \(c\).The solution to the system of equations gives:

- \(a = 0\)
- \(b = -2\)
- \(c = 5\)

Thus, the equation of the quadratic that passes through the points \((-1, 7)\), \((2, 1)\), and \((3, -1)\) is:

\[
y = -2x + 5
\]

Interestingly, this is actually a linear equation, not a quadratic one. Since \(a = 0\), the quadratic term vanishes, and the equation reduces to a linear form.

Would you like more details on the steps? Here are some related questions you can try:

1. What is the method for solving a system of three equations manually?
2. How do you know when an equation is linear or quadratic?
3. What happens if different points yield a quadratic equation?
4. Can you have more than one solution for a system like this?
5. How can you verify the solution by substituting back into the equation?

**Tip:** When solving for quadratic equations, if the highest-degree term is zero, the equation simplifies into a lower-degree polynomial.

Try to find the equation of the quadratic which passes through the points (-1,7), (2,1), and (3,-1). Explain your answer.

Learn how to find the equation of a quadratic that passes through the points (-1,7), (2,1), and (3,-1) with a detailed step-by-step solution. This problem covers key algebraic concepts, including forming and solving a system of linear equations, and demonstrates how a quadratic equation can simplify into a linear equation.

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