The solution of 4x+1 + 41−x = 10 is

Concept

An exponential equation can be simplified by replacing a positive repeated power with one variable.

For y = 4x, the laws 4x+1 = 4y and 41-x = 4/y convert the equation into a quadratic in y; every positive root y is then converted back to x.

Application

1. Let y = 4x. Since 4x is positive, y is positive.

2. Rewrite 4x+1 + 41-x = 10 as 4y + 4/y = 10.

3. Multiply by y: 4y2 - 10y + 4 = 0, hence 2y2 - 5y + 2 = 0.

4. Factor: (2y - 1)(y - 2) = 0, so y = 1/2 or y = 2.

5. Since y = 4x, 4x = 1/2 gives x = -1/2 and 4x = 2 gives x = 1/2.

Cross-check

For x = -1/2, the two terms are 41/2 = 2 and 43/2 = 8; for x = 1/2, they are 43/2 = 8 and 41/2 = 2. Each sum is 10.

Result

Thus two listed numerical values, -1/2 and 1/2, solve the equation, so the answer is “More than one of the above.”