Relationship between roots and coefficents of quadratic equations

Relationship between roots and coefficents of quadratic equations

Understanding Quadratic Equations

  • A quadratic equation is an equation of the form (ax^2+bx+c=0), where (a), (b), and (c) are constants.

  • Quadratic equations are characterised by their highest power being 2, hence they describe curving figures when graphed.

  • The solutions of a quadratic equation can be found using either factorisation, completing the square or the quadratic formula.

The Quadratic Formula

  • The quadratic formula ((-b +- sqrt(b^2-4ac))/2a) is obtained by completing the square on the general quadratic equation.

  • The discriminant of a quadratic equation is the term under the square root in the quadratic formula: (b^2-4ac).

  • The discriminant determines the nature of the roots of a quadratic equation. If it’s positive, there are two distinct roots. If it’s zero, there are exactly two identical roots (or just one root). If it’s negative, there are no real roots.

Relationship Between Roots and Coefficients

  • For a quadratic equation (ax^2 + bx + c = 0) with roots (p) and (q), Sum of roots = (-b/a) and Product of roots = (c/a).

  • These relationships between the roots and coefficients of a quadratic equation are derived from a process called ‘comparing coefficients’ in the original equation and the equation obtained by expressing the quadratic equation as (a(x-p)(x-q) = 0).

  • The sum and product of roots are also known as the Vieta’s formulas for quadratic equations.

Implications and Applications of Root-Coefficient Relationships

  • Root-coefficient relationships are useful in finding unknowns within quadratic equations without needing to fully solve the equations.

  • These relationships are particularly useful when needing to form a quadratic equation with a known sum and/or product of roots.

Remember, understanding the nature of roots, the quadratic formula and root-coefficient relationships is fundamental in dealing with quadratic equations, a vital part of further pure mathematics. Continual practice and problem solving can enable a deeper understanding of these concepts.