Find a quadratic polynomial each with the given numbers as the zeros of the . . .

Question : Find a quadratic polynomial each with the given numbers as the zeros of the polynomials

i) 3+√7, 3-√7

ii) √3, -2√3

Doubt by Aditi

Solution : 

i) Here 

α = 3+√7
β = 3-√7

We know, the quadratic equation is given by 

p(x) = x²-(α+β)x+αβ
p(x) = x²-(3+√7+3-√7)x+(3+√7)(3-√7)
p(x) = x²-(3+3)x+[(3)²-(√7)²]
p(x) = x²-6x+[9-7]
p(x) = x²-6x+2

which is the required quadratic equation.

ii) Here

α = √3,
β = -2√3

p(x) = x²-[√3+(-2√3)]x+√3(-2√3)

p(x) = x²-[√3-2√3]x-6
p(x) = x²-[-√3]x-6

p(x) = x²+√3x-6

which is the required quadratic equation.