(x-b)%2B(x-b)(x-c)%2B(x-c)(x-a)%3D0-are-always.jpeg "(x-a)(x-b)+(x-b)(x-c)+(x-c)(x-a)=0 Are Always")
The Significance of (x-a)(x-b)+(x-b)(x-c)+(x-c)(x-a)=0
The equation (x-a)(x-b)+(x-b)(x-c)+(x-c)(x-a)=0 might look intimidating at first glance, but it holds a fascinating and important truth. This equation, when solved, reveals a crucial relationship between the variables a, b, and c. Let's delve into why this equation is always true.
Expanding the Equation
To understand the significance, let's start by expanding the equation. By distributing the terms, we get:
x² - ax - bx + ab + x² - bx - cx + bc + x² - cx - ax + ac = 0
Simplifying the Equation
Combining like terms, we can simplify the equation to:
3x² - 2(a+b+c)x + (ab + bc + ac) = 0
The Relationship between Roots and Coefficients
Now, let's examine the relationship between the roots of a quadratic equation and its coefficients. In a quadratic equation of the form ax² + bx + c = 0, the sum of the roots is given by -b/a and the product of the roots is given by c/a.
Comparing this to our simplified equation:
- Sum of roots: 2(a+b+c)/3
- Product of roots: (ab + bc + ac)/3
The Significance Unveiled
The equation (x-a)(x-b)+(x-b)(x-c)+(x-c)(x-a)=0 represents a quadratic equation where:
- The sum of the roots is (a+b+c)/3.
- The product of the roots is (ab + bc + ac)/3.
This equation, therefore, represents a quadratic equation where the roots are directly related to the variables a, b, and c. This relationship highlights a crucial principle in algebra – the connection between the roots of a polynomial and its coefficients.
Applications
Understanding the relationship between the roots and coefficients of a quadratic equation is a powerful tool in various areas of mathematics, such as:
- Solving quadratic equations: Knowing the sum and product of the roots can help in finding the solutions to a quadratic equation without resorting to the quadratic formula.
- Factorization: The relationship helps in factoring quadratic expressions and understanding the nature of their roots.
- Polynomial theory: The principle generalizes to higher-order polynomials, providing insights into the behavior of polynomial functions.
In conclusion, the equation (x-a)(x-b)+(x-b)(x-c)+(x-c)(x-a)=0 is always true because it reflects a fundamental connection between the roots and coefficients of a quadratic equation. This relationship is a cornerstone of algebraic theory and finds applications in various mathematical fields.