FORMATION OF THE QUADRATIC EQUATION FROM COMPLEX ROOTS

Form quadratic equations with the following given numbers as its roots.

Problem 1 :

3 + i , 3 – i

Solution :

α  = 3 + i and β = 3 - i

x² - 6x + 10 = 0

Form quadratic equations with the following given numbers as its roots.

Problem 2 :

4 + 5i, 4 - 5i

Solution :

α  = 4 + 5i and β = 4 - 5i

Problem 3 :

Find a quadratic polynomial whose zeroes are 2 + √3 and 2 – √3

Solution :

Here α = 2 + √3 and β = 2 – √3

α + β = 2 + √3 + 2 – √3 ==> 4

α β = (2 + √3)(2 – √3)

= 2² - √3²

= 4 - 3

= 1

x² - 4x + 1 = 0

So, the required polynomial is x² - 4x + 1 = 0.

Problem 4 :

What is an equation whose roots are 5+√2 and 5−√2

Solution :

α = 5+√2, β = 5-√2

α + β = 5 + √2 + 5 - √2

α + β = 10

αβ = (5+√2) (5-√2)

αβ = 5²-√2²

 = 25 - 2

= 23

x² - (α + β) x + α β = 0

x² - 10x + 23 = 0

Problem 5 :

Find the quadratic polynomial with rational coefficients which has 1/(3+2√2) as a root.

Solution :

x² - 6x + 1 = 0

Problem 6 :

Form quadratic equations with the following given numbers as its roots.

2 + √3, 2 - √3

Solution :

α = 2 + √3, β = 2 - √3

α + β = 2 + √3 + 2 - √3

α + β = 4

αβ = (2 + √3) (2 - √3)

αβ = 2²-√3²

 = 4 - 3

= 1

x² - (α + β) x + α β = 0

x² - 4x + 1 = 0