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If α and β are the roots of the equation x22x+3=0. Find the equation whose roots are α1α+1,β1β+1

a) 3x22x+1=0

b) x2x+3=0

c) 5x22x+3=0 

d) x22x+7=0


Answer
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Hint: For the given quadratic equation first find the sum of roots (α+β) and product of roots (αβ). Now using this find the sum of roots (α1α+1+β1β+1) and product of roots (α1α+1β1β+1) of the second quadratic equation and substitute these values in standard form of quadratic equation to find the required solution.


Complete step by step answer:

It is given that α and β are the roots of the quadratic equation x22x+3=0...............(1)

So using the concept of sum of roots α+β=ba and product of roots αβ=ca.

Comparing equation (1) with standard form of quadratic equation i.e. ax2+bx+c=0 we get a=1, b=-2 and c=3.

We get α+β=2 and αβ=3

Now we have to find equation with roots α1α+1,β1β+1

So firstly lets calculate α1α+1+β1β+1=(α1)(β+1)+(α+1)(β1)(α+1)(β+1)

Lets simplify the numerator part,

αβ+αβ1+αβα+β1αβ+α+β+1=2αβ2αβ+α+β+1

Now substituting the values of αβ and α+β

We have 623+2+1=46=23.............(2)

Now in the similar way we will be computing the value of 

(α1α+1)(β1β+1)=αβαβ+1αβ+α+β+1

Again substituting the values of α and β, we have 

32+13+2+1=26=13................(3)

Now using the concept that we can write a quadratic equation using some of roots and products of roots which is x2 - (sum of roots)x +(products of roots) = 0 

Hence the equation having roots α1α+1, β1β+1 is

x223x+13=0

On simplification we get 3x22x+1=0

Hence option (a) is the right answer.


Note - In such types of problems always use the concept of sum of roots and product of roots and use this to obtain the quadratic equation.