Relation Between Roots and Coefficients

Relation Between Roots and Coefficients of a Quadratic Equation

Quadratic equations are represented in the form:

ax2 + bx + c = 0

Where:

  • a, b, and c are coefficients.
  • x represents the variable.

1. Sum of Roots:

  • The sum of the roots (solutions) of a quadratic equation ax2 + bx + c = 0 is given by -b/a.

  • The sum of the roots is independent of the value of c.

  • This relationship can be written as: α + β = -b/a.

2. Product of Roots:

  • The product of the roots (solutions) of a quadratic equation ax2 + bx + c = 0 is given by c/a.

  • The product of the roots is independent of the value of b.

  • This relationship can be written as: αβ = c/a.

3. Vieta's Formulas:

  • The relationships between the coefficients and the roots of a quadratic equation are known as Vieta's formulas.

  • These formulas hold true for both real and complex roots.

4. Discriminant and Nature of Roots:

  • The discriminant, D, which is calculated as D = b2 - 4ac, plays a crucial role in determining the nature of the roots.

  • If D > 0, the quadratic equation has two distinct real roots.

  • If D = 0, the equation has one real root (a repeated root).

  • If D < 0, the equation has two complex (non-real) roots.

5. Using the Relations:

  • Vieta's formulas are particularly useful in various mathematical and real-life contexts.

  • They are used in solving problems involving quadratic equations, in simplifying expressions, and in finding relationships between the roots of equations.

6. Example:

  • Given a quadratic equation 2x2 - 5x + 3 = 0.

  • Using Vieta's formulas:

    • Sum of roots (α and β): α + β = -(-5)/2 = 5/2
    • Product of roots: αβ = 3/2
  • You can use these relations to find the values of α and β.