Find the exact solution for each of the following:

Question

1. Find the exact solution for each of the following: 4x^2=16^x+4, 8x^3/2+216x^-3/2=19

 

2. Evaluate each of the following: (4a/9a^-1)^-3/2 and (625/4d^2)^-1/2

 

Answer 1

1. Is there a typo in the first one? It seems kind of odd to solve for 4x² = 16^x + 4.

 

For the second one, let y = x^3/2. The equation is

 

8y + (216/y) = 19

 

Multiply both sides by y and re-arrange to turn it into a quadratic equation.

 

8y² - 19y + 216 = 0

 

The solutions are y = (19 ± sqrt[-6551]) / 16, so x = [(19 ± sqrt[-6551]) / 16]^2/3; the values are 1.88 ± 2.34i.

 

2. The first one is (4a² / 9)^-3/2 = (9 / 4a²)^3/2 = (27 / 8a³).

 

The second one is (625 / 4d²)^-1/2 = (4d² / 625)^1/2 = (2d / 25)