Prove that the roots of (x-a)(x-b)+(x-b)(x-c)+(x-c)(x-a)=0 are always real and they will be equa

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Prove that the roots of (x-a)(x-b)+(x-b)(x-c)+(x-c)(x-a)=0...

Key Concepts

Quadratic Equations

A quadratic equation is a second-degree polynomial equation in the form ax² + bx + c = 0. Its solutions, or roots, can be determined by various methods including factoring, completing the square, or using the quadratic formula. The nature of these roots—whether real or complex—depends on the coefficients and specifically on the discriminant associated with the equation.

Discriminant

The discriminant of a quadratic equation, calculated as b² - 4ac, is crucial in determining the nature of the roots. A positive discriminant indicates two distinct real roots, a zero discriminant results in a single (repeated) real root, and a negative discriminant implies two complex conjugate roots. In the context of the problem, showing that the discriminant is non-negative is key to proving that the roots are always real.

Symmetric Polynomials

Symmetric polynomials are expressions that remain invariant under any permutation of the variables. In the given problem, the structure of the expression, involving pairs of terms like (x - a)(x - b), inherently creates symmetry in the parameters a, b, and c. This symmetry can simplify the analysis and helps in deriving polynomial identities useful for understanding the relationships between the coefficients and the roots.

Repeated Roots and Equality Conditions

A repeated root in a quadratic equation occurs when the discriminant is exactly zero. This scenario corresponds to the case where the quadratic's graph touches the x-axis at a single point without crossing it. In the problem, establishing that the roots are equal if and only if a = b = c involves verifying that the conditions for a double root—the vanishing of the discriminant—are met solely when the parameters are equal, thereby linking the algebraic condition to the geometric property of the quadratic.

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