(x-c)%2B(x-c)(x-a)%2B(x-a)(x-b)%3D0.jpeg "(x-b)(x-c)+(x-c)(x-a)+(x-a)(x-b)=0")
Exploring the Equation (x-b)(x-c)+(x-c)(x-a)+(x-a)(x-b)=0
This equation, though seemingly complex, holds interesting mathematical properties. It represents a symmetric expression, meaning the variables a, b, and c can be interchanged without affecting the equation's validity. Let's delve deeper into its features and applications.
Simplifying the Equation
We can simplify the equation by expanding the products:
(x-b)(x-c)+(x-c)(x-a)+(x-a)(x-b)=0
x² - bx - cx + bc + x² - ax - cx + ac + x² - ax - bx + ab = 0
3x² - 2(a+b+c)x + (ab + ac + bc) = 0
This simplified form is a quadratic equation in terms of x.
Solving for x
We can now solve for x using the quadratic formula:
x = (-b ± √(b² - 4ac)) / 2a
where a = 3, b = -2(a+b+c), and c = (ab + ac + bc).
Geometric Interpretation
The equation has a geometric interpretation. It represents the circumcircle of a triangle, where a, b, and c are the coordinates of the vertices. The solutions for x represent the x-coordinates of the points where the circumcircle intersects the x-axis.
Applications
This equation has various applications in:
- Geometry: Finding the circumcircle of a triangle.
- Algebra: Solving for roots of a quadratic equation.
- Coordinate Geometry: Determining points of intersection between a circle and the x-axis.
Summary
The equation (x-b)(x-c)+(x-c)(x-a)+(x-a)(x-b)=0, though appearing complex, is a symmetric quadratic equation with interesting geometric and algebraic interpretations. It finds applications in various mathematical areas, demonstrating the interconnectedness of different branches of mathematics.