2.6 Quadratic Equations, SPM Practice (Paper 2)


2.6 Quadratic Equations, SPM Practice (Paper 2)
Question 2:
Given α and β are two roots of the quadratic equation (2x + 5)(x + 1) + p = 0 where αβ = 3 and p is a constant.
Find the value p, α and of β.

Solutions:
(2x + 5)(x + 1) + p = 0
2x2 + 2x + 5x + 5 + p = 0
2x2 + 7x + 5 + p = 0
*Compare with, x2 – (sum of roots)x + product of roots = 0
x 2 + 7 2 x+ 5+p 2 =0 divide both  sides with 2  
Product of roots, αβ = 3
5+p 2 =3  
5 + p = 6
p = 1

Sum of roots = 7 2  
  α+β= 7 2   (1) and αβ=3   (2) from (2), β= 3 α    (3) Substitute (3) into (1), α+ 3 α = 7 2  

2 + 6 = 7α ← (multiply both sides with 2α)
2 + 7α + 6 = 0
(2α + 3)(α + 2) = 0
2α + 3 = 0      or        α + 2 = 0
α= 3 2                       α = –2

Substitute α= 3 2  into (3), β= 3 3 2 =3( 2 3 )=2  

Substitute α = –2 into (3),
β= 3 2  p=1, and when α= 3 2 ,β=2 and α=2,β= 3 2 .