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

Problem 6 :

If one zero of the polynomial x² - 5x + 3 + √3 is 2 + √3, then the other zero is '

a) 1 - √3     b) 2 - √3      c) 3 - √3     d) 2 + √3

Solution :

x² - 5x + 3 + √3

Since one zero of the polynomial is 2 + √3, the other zero will be 2 - √3. So, option b is correct.

Problem 7 :

Find a possible pair of integer values for a and c so that the equation

ax² − 4x + c = 0

has two imaginary solutions. Then write the equation.

Solution :

ax² − 4x + c = 0

a = a, b = -4 and c = c

Since it has imaginary zeroes, b² - 4ac < 0

(-4)² - 4ac < 0

16 - 4ac < 0

-4ac < -16

Dividing by -4 on both sides, we get

ac > 4

Because ac > 4, choose two integers whose product is greater than 4, such as a = 2 and c = 3. So, one possible equation is 2x² − 4x + 3 = 0

Problem 8 :

An in-ground pond has the shape of a rectangular prism. The pond has a depth of 24 inches and a volume of 72,000 cubic inches. The length of the pond is two times its width. Find the length and width of the pond.

Solution :

Depth = 24 inches

volume = 72000

Length ⋅ width ⋅ height = 72000

Length = 2w

2w ⋅ w ⋅ 24 = 72000

2w² = 72000/24

2w² = 3000

w² = 3000/2

w² = 1500

w = √1500

w = 38.72

Approximately 38 inches

length = 2(38)

= 76 inches

So, the length and width of the pond is 76 inches and 38 inches respectively.