Algebra - Part 8 - Quadratic Equation, Discriminant, Nature of Roots

Linear equation in x

x = 3

The highest power of x in the above equation is 1. 
So it is called Linear Equation.

Quadratic equation in x

x2 + 2x + 1 = 0

In the above equation, the highest power of x is 2. 
So this equation is called Quadratic Equation.

Some other quadratic equations

x2  – 25 = 0          x2 + 5x = 0

General form of a Quadratic Equation
ax2 + bx + c = 0 where a,b,c are constants and a ¹  0.

A quadratic equation has two solutions, known as roots.

Roots of a quadratic equation may be equal or distinct (not the same value) . 
Some quadratic equations do not have real roots.

Nature of roots can be determined by the value of the discriminant, D =  b2 – 4ac.

If D > 0, the roots are distinct.

If D = 0, the roots are equal.

If D < 0, then there are no real roots.

For, x2 + 2x + 1 = 0, 

a = 1, b = 2, c = 1

D =  b2 – 4ac

D =  22 – 4(1)(1) = 0

So this equation has equal roots.

For, 2x2 + 4x – 5 = 0, 

a = 2, b = 4, c =  – 5

D =  b2 – 4ac

D =  42 – 4(2)(– 5) = 56 > 0

So the roots of this equation are distinct.