using the Quadratic y= a𝑥2 + b𝑥 + c solve with **(250, 90), (270, 110), (290, 90)**

To find the quadratic equation \( y = ax^2 + bx + c \) that passes through the points \((250, 90)\), \((270, 110)\), and \((290, 90)\), we need to set up a system of equations based on the given points.

### Step 1: Set Up the Equations
For each point \((x, y)\), substitute \(x\) and \(y\) into the quadratic equation.

1. For \((250, 90)\):
   \[
   90 = a(250)^2 + b(250) + c
   \]

2. For \((270, 110)\):
   \[
   110 = a(270)^2 + b(270) + c
   \]

3. For \((290, 90)\):
   \[
   90 = a(290)^2 + b(290) + c
   \]

### Step 2: Solve the System of Equations
Let's write these equations in a simplified form:

1. \( 62500a + 250b + c = 90 \)
2. \( 72900a + 270b + c = 110 \)
3. \( 84100a + 290b + c = 90 \)

This is a system of three linear equations in three variables \(a\), \(b\), and \(c\). We can solve this system using methods like substitution, elimination, or matrix operations. I’ll use matrix operations to find \(a\), \(b\), and \(c\).

### Step 3: Solve Using Matrix Operations
We can express this system in matrix form \( \mathbf{A}\mathbf{x} = \mathbf{y} \) where:

\[
\mathbf{A} = \begin{pmatrix}
62500 & 250 & 1 \\
72900 & 270 & 1 \\
84100 & 290 & 1 \\
\end{pmatrix},
\quad
\mathbf{x} = \begin{pmatrix}
a \\
b \\
c \\
\end{pmatrix},
\quad
\mathbf{y} = \begin{pmatrix}
90 \\
110 \\
90 \\
\end{pmatrix}
\]

Let's solve for \(\mathbf{x}\).The solution for the quadratic equation \( y = ax^2 + bx + c \) based on the given points is:

\[
a = -0.05, \quad b = 27, \quad c = -3535
\]

Thus, the quadratic equation is:

\[
y = -0.05x^2 + 27x - 3535
\]

Would you like any details on how this was calculated or have any questions?

Here are five related questions:
1. How do you find the vertex of the quadratic equation?
2. Can you calculate the roots of the quadratic equation \( y = -0.05x^2 + 27x - 3535 \)?
3. What is the axis of symmetry for this quadratic equation?
4. How would you determine the maximum or minimum value of this quadratic function?
5. How can you verify that these points actually lie on the calculated quadratic curve?

**Tip:** When solving a system of equations, using matrix methods like Gaussian elimination or Cramer's rule can be very efficient for larger systems.

Learn how to find the quadratic equation y = ax^2 + bx + c that passes through the points (250, 90), (270, 110), and (290, 90). This problem involves solving a system of equations using matrix operations and is suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation