2^(4x-1) -4^(x+1/2)+2=0 Solve.

Question

Struggling with these!

Answer 1

2^4x-1 - 4^x+½ + 2 = 0

 

2^4x2^-1 - 4^x4^½ + 2 = 0

 

½(2^2x)² - 2(4^x) + 2 = 0

 

½(4^x)² - 2(4^x) + 2 = 0

 

(4^x)² - 4(4^x) + 4 = 0

 

Let u = 4^x   Then u² - 4u + 4 = 0

 

                        (u-2)² = 0

 

                              u = 2

 

4^x = 2    So, (2²)^x = 2¹

 

                   2^2x = 2¹

 

                    2x = 1     x = 1/2

 

Answer 2

Get common base of 2.

 

2^{(4x - 1)} - (2²)^{(x + 1/2)} + 2 = 0

 

 

Apply rules of exponents.

 

2^{(4x - 1)} - 2^{(2x + 1)} + 2 = 0

 

(2^x)⁴(1/2) - (2^x)²(2) + 2 = 0

 

 

Let         q = 2^x

 

(1/2)q⁴ - 2q² + 2 = 0

 

 

Multiply both sides of the equation by 2.

 

q⁴ - 4q² + 4 = 0

 

 

Factor the left side using FOIL.

 

(q² - 2)(q² - 2) = 0

 

(q - √2)(q + √2)(q - √2)(q + √2) = 0

 

 

Set the factors equal to zero to solve for q.  Then substitute the values to find x.

 

x = log(q) / log(2)