Question

Write a quadratic polynomial whose zeroes are √5+2 and √5-2

Answer 1

Answer :

The required quadratic polynomial is

x² - 2√5x + 1

Given :

The zeroes of a quadratic polynomial are :

• (√5 + 2) and (√5 - 2)

To Find :

• The quadratic polynomial

Formula to be used :

If sum and product of zeroes of a polynomial are given then the polynomial can be written as :

• x² - (sum of the zeroes)x + product of the zeroes

Solution :

Given , zeroes :

(√5 + 2) and (√5 - 2)

Sum of the zeroes = √5 +2 + √5 - 2

→ Sum of the zeroes = 2√5

And

product of the zeroes = (√5 + 2)(√5 - 2)

→ product of the zeroes = (√5)² - 2²

→ product of the zeroes =5 - 4

→ product of the zeroes = 1

Therefore , the quadratic polynomial is :

x² - 2√5 x + 1

Answer 2

Given

• Roots of required polynomial :

√5 + 2 & √5 - 2

To find:

• Quadratic polynomial

Solution:

We know that

Quadratic polynomial : x² - (sum of roots)x + product of roots

Finding sum of roots & Product of roots

→ Sum of roots : √5 + 2 + √5 - 2

→ Sum of roots : 2√5

$\underline{\rule{259}{2}}$

★ Product of roots : (√5 + 2)(√5 - 2)

Using

(a + b)(a - b) = a² - b²

★ Product of roots : (√5)² - 2²

★ Product of roots : 5 - 4

★ Product of roots : 1

Now ,

Quadratic polynomial : x² - 2√5x + 1

Hence required quadratic polynomial is : x² - 2√5 + 1