Factorization of Quadratic Equations – SS1 Mathematics Lesson Note (First Term)

Construction of quadratic equations from sum and product of roots

We can find the sum and product of the roots directly from the coefficient in the equation. It is usual to call the roots of the equation α and β  If the equation 

ax² +bx + C = 0   …………….  i

has the roots α and β then it is equivalent to the equation

(x – α  )( x – β )  = 0

x² – βx –  βx + αβ  = 0 ………… ii

Divide equation (i)by the coefficient  of x²

ax²+ bx + C     = 0  …………   ii

 a       a      a

Comparing equations (2) and (3)

x2+  bx  +  C     = 0

         a       a

x2  –  ( α +β)x  + αβ    = 0

then

α+ β= -b

            a

and αβ = C

                 a

For any quadratic equation, ax² +bx + C = 0 with roots α and β

α + β  = -b

               a

αβ =  C

          a

Examples

1. If the roots of 3×2– 4x – 1 = 0 are αand β, find  α + β and αβ
2. if α and βare the roots of the equation 

     3x² – 4x – 1 = 0 , find the value of 

(a)     α     +  β

         β          α

(b)   α   –   β

Solutions

1. Since α + β =  -b

                              a

Comparing the given equation 3x² – 4x – 1= 0 with the general form

ax² + bx + C = 0

a = 3, b = -4, C = 1.

Then 

α + β = -b  =  –(-4)

              a           3

        = + 4  =  +1 1/3

              3

αβ =C   =  -1    = -1

        a         3         3

2.aα +  β    =  α2 +β2

    β     α               αβ

= (α + β )2 – 2αβ

            αβ

Here, comparing the given equation, with the general equation,

a = 3, b = -4, C = – 1

from the solution of Example 1 (since the given equation is the same ),

α + β = -b =  – (-4)   = +4

                           3           3

αβ  = C   = – 1

          a         3

then

α  +    β    =  ( α+ β ) 2 – 2 αβ

β         α                  αβ

=  (4/3 ).2 – 2 ( – 1/3 )

              1/3

=   16        ± 2

   9           3

– 1

            3

=   16 + 6   ÷    -1/3

         9

22     x   -3

         9          1

=   –22

       3

1. b)  Since 

(α-β) 2 =α2 + β- 2 α β

but

α2 + β2 = ( α + β)2 -2 α β

:.(α- β)2 = ( α+ β )2 – 2αβ -2αβ

(α – β)2 = (α + β )2 – 4α β

:.( α – β) = √(α + β )2 – 4αβ

( α – β) =√   (4/3 )2 – 4 ( – 1/3 )

= √ 16/9  +4/3

 =    √16 + 12

              9

  =      √28           =   √28

            9                    3

    :. α – β  =     √28 

                             3