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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.
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.
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
)(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.
)(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.
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