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If α and β are the zeroes of the quadratic polynomial $x^{2} + 10 x + 30$ then find the quadratic polynomial whose zeroes are $α + 2 β$ and $2 α + β$ .

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x^{2} + 30 x + 230

x^{2} + 30 x - 230

x^{2} - 30 x + 230

x^{2} - 30 x - 230

answer is A.

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Detailed Solution

Given, α and β are the zeroes of the quadratic polynomial

x^{2} + 10 x + 30

.
We know that, the relationship between zeroes and coefficient of quadratic polynomial is

Sum of Zeroes = - \frac{Coefficient of x}{Coefficient of x^{2}}

and

Product of Zeroes = \frac{Constant Term}{Coefficient of x^{2}}

.
Then, we get,

α + β = - \frac{10}{1} = - 10

and

αβ = \frac{30}{1} = 30

… (1)
We have to find out quadratic polynomial whose zeroes are

α + 2 β

and

2 α + β

.
Let

α^{'} and β'

are the new zeroes then,

α^{'} = α + 2 β

and

β^{'} = 2 α + β

.
Then,

α^{'} + β^{'} = α + 2 β + 2 α + β

\Rightarrow α^{'} + β^{'} = 3 α + 3 β

\Rightarrow α^{'} + β^{'} = 3 (α + β)

\Rightarrow α^{'} + β^{'} = 3 (- 10)

{using equation (1)}

\Rightarrow α^{'} + β^{'} = - 30

… (2)
And,

α^{'} β^{'} = (α + 2 β) (2 α + β)

\Rightarrow α^{'} β^{'} = {2 α}^{2} + {2 β}^{2} + 5 αβ

\Rightarrow α^{'} β^{'} = {2 (α}^{2} + β^{2}) + 5 \times 30

{using equation (1)}

\Rightarrow α^{'} β^{'} = {2 (α}^{2} + β^{2}) + 150

… (3)
Now,

α + β = - 10

. On squaring both sides,

{(α + β)}^{2} = {(- 10)}^{2}

{

{(a + b)}^{2} = a^{2} + b^{2} + 2 ab

}

\Rightarrow α^{2} + β^{2} + 2 αβ = 100

\Rightarrow α^{2} + β^{2} + 2 \times (30) = 100

{using equation (1)}

\Rightarrow α^{2} + β^{2} + 60 = 100

\Rightarrow α^{2} + β^{2} = 40

Using above value in equation (3),

α^{'} β^{'} = 2 \times 40 + 150 = 80 + 150 = 230

… (4)
Then,
Quadratic Polynomial

= x^{2} - (Sum of zeroes) x + (Product of zeroes)

Quadratic Polynomial

= x^{2} - (α^{'} + β^{'}) x + α^{'} β^{'}

Quadratic Polynomial

= x^{2} - (- 30) x + 230

Quadratic Polynomial

= x^{2} + 30 x + 230

Therefore, the quadratic polynomial whose zeroes are

α + 2 β

and

2 α + β

x^{2} + 30 x + 230

.
Hence, option (1) is correct.

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If α and β are the zeroes of the quadratic polynomial x2+10x+30 then find the quadratic polynomial whose zeroes are α+2β and 2α+ β.

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If α and β are the zeroes of the quadratic polynomial $x^{2} + 10 x + 30$ then find the quadratic polynomial whose zeroes are $α + 2 β$ and $2 α + β$ .