Question

solve the following equations by reducing them to quadratic equations​

Answer 1

Answer:

n = 0 , 2

Solution:

Here ,

The given equation is ;

4ⁿ - 5•2ⁿ + 4 = 0

The given equation can be rewritten as ;

=> (2²)ⁿ - 5•2ⁿ + 4 = 0

=> 2²ⁿ - 5•2ⁿ + 4 = 0

=> (2ⁿ)² - 5•2ⁿ + 4 = 0 -------(1)

Substituting 2ⁿ = y in eq-(1) , it will be reduced to a quadratic equation.

Thus,

=> y² - 5y + 4 = 0

=> y² - 4y - y + 4 = 0

=> y(y - 4) - (y - 4) = 0

=> (y - 4)(y - 1) = 0

=> y = 1 , 4

Case(1) : If y = 1

=> y = 1

=> 2ⁿ = 1

=> 2ⁿ = 2°

=> n = 0

Case(2) : If y = 4

=> y = 4

=> 2ⁿ = 4

=> 2ⁿ = 2²

=> n = 2

Hence,

Required solutions are :

n = 0 , 2