Question

Solve the following quadratic equation:

`4^((x + 1)) + 4^((1 - x)) = 10`

Answer 1

Given:

`4^((x + 1)) + 4^((1 - x)) = 10` 

⇒ `4^(x + 1) + 4 · 4^x, 4^(1 - x) = 4/(4^x)`

⇒ `4 · 4^x + 4/(4^x) = 10`

Let 4^x be y. 

∴ `4y + 4/y = 10` 

⇒ 4y² – 10y + 4 = 0 

⇒ 4y² – 8y – 2y + 4 = 0 

⇒ 4y(y – 2) – 2(y – 2) = 0 

⇒ `y = 2` or `y = 2/4 = 1/2` 

⇒ `4^x = 2` or `1/2` 

⇒ 4^x = 2^2x = 2¹ or 2^2x = 2^–1 

⇒ `x = 1/2` or `x = (-1)/2` 

Hence, `1/2`  and `(-1)/2` are roots of the given equation.