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
x2+72x+5+p2=0←divide both sides with 2
Product of roots, αβ = 3
5+p2=3
5 + p = 6
p = 1
Sum of roots =
−72
α+β=−72 → (1)and αβ=3 → (2)from (2), β=3α → (3)Substitute (3) into (1),α+3α=−72
2α2+ 6 = –7α ← (multiply both sides with 2α)
2α2+ 7α + 6 = 0
(2α + 3)(α + 2) = 0
2α + 3 = 0 or α + 2 = 0
α=− 3 2 α = –2
Substitute α=−32 into (3),β=3−32=3(−23)=−2
Substitute α = –2 into (3),
β=−32∴