# Introduction To Quadratic Equations

INTRODUCTION:

An equation in one unknown quantity (let it be x) in the form

**ax**is known as a quadratic equation, where^{2}+ bx + c = 0**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 ax

^{2}+ bx + c = 0, then (x - ) is a factor of the expression ax^{2}+ bx + c = 0; on the other hand, if (x - ) is a factor of ax^{2}+ bx + c = 0 then is a root of the equation ax^{2}+ bx + c = 0.
Proof:

Given that is a root of the quadratic equation

ax

Hence, a

Now, ax

= a(x

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

Which clearly shows that (x - ) is a factor of the expression ax

On the other hand, if (x - ) is a factor of ax

ax

where p (≠ 0) and q are constants.

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

a

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

Hence, is a root of the quadratic equation ax

ax

^{2}+ bx + c = 0 (1)Hence, a

^{2}+ b + c = 0Now, ax

^{2}+ bx + c = ax^{2}+ bx + c - (a^{2}+ b + c) [Since a^{2}+ b + c = 0]= a(x

^{2}-^{2}) + b(x - )= (x - )[a(x + )+b]

Which clearly shows that (x - ) is a factor of the expression ax

^{2}+ bx + c.On the other hand, if (x - ) is a factor of ax

^{2}+ bx + c then,ax

^{2}+ 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) = 0which shows that the equation (1) is satisfied by x = .

Hence, is a root of the quadratic equation ax

^{2}+ bx + c = 0.
THEOREM 2:

A quadratic equation has two and only two roots.

Proof:

Let ax

^{2}+ 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.

**Other topics under Quadratic Equations:**