- #1

RChristenk

- 64

- 9

- Homework Statement
- Solve ##16x^4=81##

- Relevant Equations
- Algebra

##x^4=\dfrac{81}{16}##

##x=\pm\dfrac{3}{2}##.

But I recently realized there are complex solutions as well:

##16x^4-81=0##

##(4x^2)^2-9^2=0##

##(4x^2+9)(4x^2-9)=0##

##x^2=\dfrac{-9}{4}, x^2=\dfrac{9}{4}##

##x=\pm\dfrac{3i}{2}, x=\pm\dfrac{3}{2}##

Intuitively when I see ##16x^4=81##, I see a straightforward solution of ##x=\pm\dfrac{3}{2}##. But solving problems in this way clearly excludes the complex solutions. Why is that so? Does it mean this straightforward approach to solving these kinds of problems is wrong?

##x=\pm\dfrac{3}{2}##.

But I recently realized there are complex solutions as well:

##16x^4-81=0##

##(4x^2)^2-9^2=0##

##(4x^2+9)(4x^2-9)=0##

##x^2=\dfrac{-9}{4}, x^2=\dfrac{9}{4}##

##x=\pm\dfrac{3i}{2}, x=\pm\dfrac{3}{2}##

Intuitively when I see ##16x^4=81##, I see a straightforward solution of ##x=\pm\dfrac{3}{2}##. But solving problems in this way clearly excludes the complex solutions. Why is that so? Does it mean this straightforward approach to solving these kinds of problems is wrong?