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 (nonreal) roots.
5. Using the Relations:

Vieta's formulas are particularly useful in various mathematical and reallife 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.