Question

If the two zeroes of a quadratic polynomial are `±  sqrt5` then the quadratic polynomial is ______.

Options

• x² + 5

• `(x + sqrt5)^2`

• 4(x² - 5)

• `x^2 - sqrt5`

Answer 1

If the two zeroes of a quadratic polynomial are `±  sqrt5` then the quadratic polynomial is 4(x² − 5).

Explanation:

Given, zeroes of a quadratic polynomial are `±  sqrt5`.

Let α and β be the two zeroes of the given quadratic polynomial.

So, α = `sqrt5` and β = `-sqrt5`

∴ Sum of zeroes = α + β

= `sqrt5 + (-sqrt5)`

= 0

∴ Product of zeroes = αβ

= `(sqrt5)(-sqrt5)`

= −5

The equation of quadratic polynomial is

f(x) = k[x² − (α + β)x + αβ], k ∈ R

= [x² − (0)x + (−5)]

= [x² − 5]

For k = 4, f(x) = 4(x² − 5)

Hence, the quadratic polynomial is 4(x² − 5).