https://file.aiquickdraw.com/mathbot/file/1727407384492ll2kv950.jpg

The image shows three quadratic equations in two variables \(x\) and \(y\). Let's rewrite and solve each equation one by one.

1. **Equation 1**:
   \[
   x^2 + y^2 - 14x + 16y + 88 = 0
   \]
   This can be solved by completing the square for both \(x\) and \(y\).

2. **Equation 2**:
   \[
   x^2 + y^2 - 26x + 6y + 163 = 0
   \]
   Again, we'll complete the square for both \(x\) and \(y\) to find the standard form of the circle or determine the center and radius.

3. **Equation 3**:
   \[
   x^2 + y^2 - 28x + 2y + 193 = 0
   \]
   We'll follow the same procedure of completing the square.

Let me solve these equations by completing the square for all three. I'll start now.Here are the solutions after completing the square for each equation:

1. **Equation 1:**
   \[
   y = -\sqrt{-(x - 12)(x - 2)} - 8 \quad \text{or} \quad y = \sqrt{-(x - 12)(x - 2)} - 8
   \]
   
2. **Equation 2:**
   \[
   y = -\sqrt{-x^2 + 26x - 154} - 3 \quad \text{or} \quad y = \sqrt{-x^2 + 26x - 154} - 3
   \]
   
3. **Equation 3:**
   \[
   y = -\sqrt{-(x - 16)(x - 12)} - 1 \quad \text{or} \quad y = \sqrt{-(x - 16)(x - 12)} - 1
   \]

These represent the curves defined by the quadratic forms of \(x\) and \(y\).

Would you like more detailed explanations on how to interpret these curves, or any additional questions?

Here are some related questions:

1. What is the geometric meaning of completing the square for these equations?
2. How can we determine the center and radius of the circles formed by the equations?
3. What happens if the terms inside the square roots become negative?
4. How do these equations relate to conic sections?
5. Can we simplify these equations further for specific values of \(x\)?

**Tip:** When solving quadratic equations in two variables, completing the square is a powerful technique to reveal the geometric nature of the equation (e.g., circles, ellipses).

Solve the three quadratic equations involving x^2 + y^2 and other linear terms in x and y.

Learn how to solve quadratic equations involving x^2 and y^2 with linear terms in x and y using the method of completing the square. These problems demonstrate the relationship between quadratic equations and circles. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation