Question

Α and β are zeroes of the quadratic polynomial x2 – 6x + y. find the value of ‘y’ if 3α + 2β = 20.

Answer 1

Step-by-step explanation:

Given -

• p(x) = x² - 6x + y
• 3α + 2β = 20

To Find -

• Value of y

Now,

As we know that :-

• α + β = -b/a

→ α + β = -(-6)/1

→ α + β = 6

And

• αβ = c/a

→ αβ = y/1

→ αβ = y

Now,

3α + 2β = 20 can be written as :-

→ (3 + 2)(α + β) - 3β - 2α = 20

→ (3 + 2)(6) - 3β - 2α = 20

→ 5×6 - 20 = 3β + 2α

→ 30 - 20 = 3β + 2α

→ 3β + 2α = 10

Now,

→ [ 3α + 2β = 20 ] × 2

[ 2α + 3β = 10 ] × 3

→ 6α + 4β = 40

6α + 9β = 30

(-) (-) (-)

___________

→ -5β = 10

→ β = -2

Now,

→ 3α + 2β = 20

→ 3α = 20 - 2(-2)

→ 3α = 20 + 4

→ 3α = 24

→ α = 8

Henec,

The value of α is 8 and β is -2.

Now,

The value of y is

→ αβ = y

→ 8 × -2 = y

→ y = -16

Hence,

The value of y is -16.

Answer 2

Answer:

Step-by-step explanation:

Given :

α , β are zeroes of the quadratic polynomial x² - 6x + y   &

3α + 2β = 20 ... (1)

To Find :

Value of y

Solution :

Compare given polynomial x²-6x+y with ax²+bx+c , we get ,

⇒ a = 1 , b = -6 , c = y

Now ,

Sum of zeroes , α + β = -b/a = -(-6)/1

⇒ α + β = 6

⇒ 2α + 2β = 12 ... (2) [ Multiply 2 on both sides ]

Product of zeroes , αβ = c/a = y/1

⇒ αβ = y ... (3)

Now solve (2) - (1) , we get ,

⇒ 2α + 2β - 3α - 2β = 12 - 20

⇒ -α = -8

⇒ α = 8

sub. α value in (1) , we get ,

⇒ 2β = 20 - 3(8)

⇒ 2β = -4

⇒ β = -2

Now , sub. α , β values in (3) , we get ,

⇒ αβ = y

⇒ 8(-2) = y

⇒ y = -16

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