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Introduction To Quadratic Equations

Introduction Quadratic Equation Online Tutoring / Homework Help
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 Alpha be a root of the quadratic equation ax2 + bx + c = 0, then (x - Alpha) is a factor of the expression ax2 + bx + c = 0; on the other hand, if (x - Alpha) is a factor of ax2 + bx + c = 0 then Alpha is a root of the equation ax2 + bx + c = 0.

Proof:

Given that Alpha is a root of the quadratic equation

ax2 + bx + c = 0             (1)

Hence, aAlpha 2 + bAlpha + c = 0

Now, ax2 + bx + c = ax2 + bx + c - (aAlpha 2 + bAlpha + c) [Since aAlpha 2 + bAlpha + c = 0]

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

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

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

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

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

where p (≠ 0) and q are constants.

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

aAlpha 2 + bAlpha + c = 0.(px + q) = 0

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

Hence, Alpha 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


quadratic equation

quadratic equation

quadratic equation

quadratic equation

quadratic equation

Or (x - Alpha)(x - β) = 0


quadratic equation

Since equation (1) reduces to (x - Alpha)(x - β) = 0 and this equation is satisfied only by the values x = Alpha 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.