
solve using quadratic formula y=ax^2+bx+c **(80, 140), (120, 180), (80, 140)**




It seems there might be an error in the given points. You listed the point \((80, 140)\) twice. If this is intentional, please clarify; otherwise, I'll assume the second point should be different. However, I will proceed under the assumption that the third point is meant to be different, and we'll consider a scenario with two distinct points, \((80, 140)\) and \((120, 180)\).

Let's go ahead and solve this assuming that the correct points are \((80, 140)\), \((120, 180)\), and the third distinct point needs to be provided. For now, I'll proceed with just the first two points.

### Step 1: Set Up the Equations
Given the two points \((80, 140)\) and \((120, 180)\), we substitute them into the quadratic equation \( y = ax^2 + bx + c \):

1. For \((80, 140)\):
   \[
   140 = a(80)^2 + b(80) + c
   \]
   \[
   140 = 6400a + 80b + c \quad \text{(Equation 1)}
   \]

2. For \((120, 180)\):
   \[
   180 = a(120)^2 + b(120) + c
   \]
   \[
   180 = 14400a + 120b + c \quad \text{(Equation 2)}
   \]

### Step 2: Solve the System of Equations
Subtract Equation 1 from Equation 2 to eliminate \( c \):

\[
(14400a + 120b + c) - (6400a + 80b + c) = 180 - 140
\]
\[
8000a + 40b = 40
\]
\[
200a + b = 1 \quad \text{(Equation 3)}
\]

### Step 3: Find \( a \) and \( b \)
Since there are only two equations, there are multiple possible solutions for \( a \) and \( b \). If a third point is provided, the coefficients can be uniquely determined. For now, we assume the third equation is not available.

Given \( 200a + b = 1 \):

We could express \( b \) in terms of \( a \) as:
\[
b = 1 - 200a
\]

### Step 4: Assume a Value for \( c \)
If we assume \( c \), we could solve for specific values of \( a \) and \( b \).

However, without a third point, this problem is underdetermined, meaning that there is an infinite number of parabolas passing through any two given points. Typically, a third distinct point is necessary to fully determine the coefficients \( a \), \( b \), and \( c \).

### Conclusion
The equation has infinite solutions without the third point, as \( a \) and \( b \) depend on the choice of \( c \). Please provide a third distinct point to proceed further.

Would you like to clarify or provide a different third point for solving this?

### Related Questions
1. What happens to the quadratic equation if two points are identical?
2. How would adding a third point change the system of equations?
3. Can we determine a general solution with only two points?
4. What are the possible values of \( a \) and \( b \) if \( c \) is assumed to be zero?
5. How can we use the quadratic formula to find roots once \( a \), \( b \), and \( c \) are determined?

### Tip
To fully determine a quadratic function, three distinct points are required. If only two points are given, additional information or assumptions are necessary to find a unique solution.

Learn how to solve the quadratic equation y = ax^2 + bx + c with given points (80, 140) and (120, 180). Understand the process of setting up equations, solving a system of equations, and determining coefficients a, b, and c. Suitable for high school students.

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