Question

p(x) has two zeros α and β. find p(x) if α + β = 19 and α - β = 5

Answer 1

Let the quadratic equation in the form = ax²+bx+c

α+β = -b/a = 19

α-β= 5

Squaring both sides

(α+β) - 4αβ =5

19-4αβ = 5

-4αβ = 5-19

αβ = 14/4 = 7/2 = c/a

a = 2
b = -38
c = 7

Putting the value

2x²-38x+7

Answer 2

α+β=19(1)        α-β=5(2)
                         α=5+β(3)
Putting (3) into (1).
5+β+β=19
2β=19-5
2β=14
β=14/2=7←

Putting value of β in (1)
α+7=19
α=19-7
α=12←

Now, α+β=19
         αβ=12×7=84
polynomial= x²-(α+β)x+αβ
                   p(x)=²-19x+84