When does $9^x - 3^{x+1} = k$ have one or more real solutions?

Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

The equation \[9^x - 3^{x+1} = k\] has one or more real solutions precisely when

1. \(k \geq -9/4\),

2. \(k > 0\),

3. \(k \leq -1\),

4. \(k \geq 5/8\).

Maybe we can change variable so that we are working with an equation of a more familiar form? Would the substitution \(y = 3^x\) help?