Question

How do we derive the relationship between zeroes of a quadratic polynomial and its co efficients using ONLY the general form

Answer 1

Relationship between zeroes and coefficients of a quadratic polynomial

In general if α, β are the zeroes of a quadratic polynomial p( x ) = ax² + bx + c, where a ≠ 0 then ( x - α ) and ( x - β ) are the factors p( x )

So, the polynomial would be :

= ( x - α )( x - β )

where k is constant

= x( x - β ) - α( x - β )

= x² - βx - αx + αβ

= x² - ( α + β )x + αβ

Also Quadratic polynomial is ax² + bx + c

Equating both

=> x² - ( α + β )x + αβ + αβ = ax² + bx + c

Comparing on both sides

• 1 = a --- Eq( 1 )

• - ( α + β ) = b ---- Eq( 2 )

• αβ = c --- Eq( 3 )

Dividing Eq( 2 ) by ( 1 )

=> - ( α + β ) / 1 = b / a

=> - ( α + β ) = b / a

=> α + β = - b / a

Dividing Eq( 3 ) by ( 1 )

=> αβ / 1 = c / a

=> αβ = c / a

Hence derived.