2.6 Quadratic Equations, SPM Practice (Paper 2)


2.6 Quadratic Equations, SPM Practice (Paper 2)
Question 3:
If α and β are the roots of the quadratic equation 3x2 + 2x – 5 = 0, form the quadratic equations that have the following roots.
(a)  2 α  and  2 β (b) ( α+ 2 β ) and ( β+ 2 α )  

Solutions:
3x2 + 2x – 5 = 0
a = 3, b = 2, c = –5
The roots are α and β.
α+β= b a = 2 3 αβ= c a = 5 3  

(a)
The new roots are  2 α and 2 β . Sum of new roots = 2 α + 2 β = 2β+2α αβ                  = 2( α+β ) αβ = 2( 2 3 ) 5 3 = 4 5  

Product of new roots =( 2 α )( 2 β )= 4 αβ                  = 4 5 3 = 12 5  

Using the formula, x2 – (sum of roots)x + product of roots = 0
The new quadratic equation is
x 2 ( 4 5 )x+( 12 5 )=0  
5x2 – 4x – 12 = 0

(b)
The new roots are ( α+ 2 β )and( β+ 2 α ). Sum of new roots =( α+ 2 β )+( β+ 2 α )  
=α+β+( 2 α + 2 β )=α+β+ 2α+2β αβ =α+β+ 2( α+β ) αβ = 4 5 + 2( 4 5 ) 12 5 = 4 5 2 3 = 2 15

Product of new roots =( α+ 2 β )( β+ 2 α ) =αβ+2+2+ 4 αβ
= 12 5 +4+ 4 12 5 = 12 5 +4 5 3 = 1 15

The new quadratic equation is
x 2 ( 2 15 )x+( 1 15 )=0
15x2 – 2x – 1 = 0