(Discriminant of Quadratic Equation and Relation between Roots of the Quadratic Equation)

The Standard form or General Form of Quadratic Polynomial is ax2 + bx + c, where a, b and c are constants and x is the variable.

The Standard Form or General Form of Quadratic Equation is

ax2 + bx + c = 0

where a, b and c are constants and x is the variable.

The values of x which satisfies the Quadratic Equation are called its solutions or roots. A Quadratic Equation has two distinct roots or two equal roots. The roots of the equations may be Real Roots representing real numbers as well as Imaginary Roots representing Complex Numbers.

### The roots of the Quadratic Equation are ax2 + bx + c = 0 are

Now, sum of roots is

#### Quadratic equation class 10:- Hence the sum of the roots of the quadratic equation ax2 + bx + c = 0 is

Again, the roots of the Quadratic Equation are ax2 + bx + c = 0 are

### Basic Proportionality Theorem or Thales Theorem

Now, Product of roots is

Hence the products of the roots of the quadratic equation ax2 + bx + c = 0

Once again, we have, the Standard Form or General Form of Quadratic Equation is

ax2 + bx + c = 0

Dividing both sides by a, we get

The Quantity b2 – 4ac is called the Discriminant of the Quadratic Equation ax2 + bx + c = 0 and it is denoted by D.

D = b2 – 4ac

1. If Discriminant D ≥ 0, then the Quadratic Equation has two real roots.

(I) If Discriminant D > 0, the Quadratic Equation has two Distinct Real Roots.

(II) If Discriminant D = 0, the Quadratic Equation has two Equal Real Roots.

2. If Discriminant D < 0, the Quadratic Equation has NO Real Roots. It has Imaginary Roots representing Complex Numbers.