INTRODUCTION:

An equation in one unknown quantity (let it be x) in the form ax2 + bx + c = 0 is known as a quadratic equation, where a, b, c are constants and a ≠ 0 while b and c may be zero. Here a is called the coefficient of x2, b is the coefficient of x and c is a constant term. The values of x which satisfy the equation are named as the roots of the quadratic equation.

THEOREM 1:

If be a root of the quadratic equation ax2 + bx + c = 0, then (x - ) is a factor of the expression ax2 + bx + c = 0; on the other hand, if (x - ) is a factor of ax2 + bx + c = 0 then is a root of the equation ax2 + bx + c = 0.

Proof:

Given that is a root of the quadratic equation

ax2 + bx + c = 0             (1)

Hence, a 2 + b + c = 0

Now, ax2 + bx + c = ax2 + bx + c - (a 2 + b + c) [Since a 2 + b + c = 0]

= a(x2 - 2) + b(x - )

= (x - )[a(x + )+b]

Which clearly shows that (x - ) is a factor of the expression ax2 + bx + c.

On the other hand, if (x - ) is a factor of ax2 + bx + c then,

ax2 + bx + c = (x - )(px + q)             (2)

where p (≠ 0) and q are constants.

Now, putting x = on both sides of the identity (2) we get,

a 2 + b + c = 0.(px + q) = 0

which shows that the equation (1) is satisfied by x = .

Hence, is a root of the quadratic equation ax2 + bx + c = 0.

THEOREM 2:

A quadratic equation has two and only two roots.

Proof:

Let ax2 + bx + c = 0                 (1) be a quadratic equation

Or (x - )(x - β) = 0

Since equation (1) reduces to (x - )(x - β) = 0 and this equation is satisfied only by the values x = and x = β and by no other values of x, it is clear that the equation (1) has two and only two roots.

Hence a quadratic equation has two and only two roots.