Question

Form the quadratic equation whose roots are:
`2 + sqrt(5) and 2 - sqrt(5)`.

Answer 1

Let a, β be the given roots.
Then a = `2 + sqrt(5) and beta = 2 - sqrt(5)`
a + β = `2 + sqrt(5) + 2 - sqrt(5) = 4`
and aβ = `(2 + sqrt(5)) (2 - sqrt(5))`
⇒ a + β = 4 and aβ = (2)² - `(sqrt(5))^2`
⇒ a + β = 4 and aβ = 4 - 5
⇒ a + β = 4 and aβ = -1
Required quadratic equation
x² - (a + β)x + aβ = 0
⇒ x² - 4x - 1 = 0.