MathematicsIf α,β are the roots of the equation x2−3x + 1 = 0,then the equation with roots 1/(α−2) ,1/(β−2) is:

If α,β are the roots of the equation x2−3x + 1 = 0,then the equation with roots 1/(α−2) ,1/(β−2) is:


  1. A
    x2-x-1
  2. B
    2x2-x-1
  3. C
    x2-2x-1
  4. D
    x2-x-2 

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    Solution:

    Concept- Once we know a quadratic equation's two roots, we can move on to write its general formulation. The sum and product of roots for the given quadratic equation will then be determined. Finally, we must include these numbers in a new quadratic equation's simplified expression.
    Given a quadratic equation is x2-3+ 1 = 0 and α, and β are the roots of this equation.
    The quadratic equation with roots 1(α-2) ,1(β-2)  is given by:
    (x – 1) (α-2) ) (x – 1) (β-2) ) =(x(α-2) -1) (α-2) ×(x(β - 2) - 1) (β-2) 
    =x2(αβ-2(α+β) +4-x(α+β-4) +1αβ-2(α+β) +4 .
    The above expression is in the form of α+β and αβ,α+β and αβ.
    For quadratic equation x2−3x + 1 =0:
    Sum of roots
    = α+β=-ba=-(-3) 1=3
    And product of roots =αβ=ca=11=1
    Putting the values of α+β  and αβ,α+β and αβ in above equation, we get:
    =-x2++ 1-1 =x2 -- 1
    Hence, the answer is option 1.
     
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