# Relation Between Roots and Coefficients

Relation Between Roots and Coefficients of a Quadratic Equation

Quadratic equations are represented in the form:

a${x}^{2}$ + 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 a${x}^{2}$ + 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 a${x}^{2}$ + 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 = ${b}^{2}$ - 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 2${x}^{2}$ - 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.