Challenge Level

Find all real solutions to this equation:

$$\left(2-x^2\right)^{x^2-3\sqrt{2}x+4} = 1$$

Extension: What if $x$ is permitted to be a complex number?

Did you know ... ?

Quadratic equations and powers are commonly used throughout school and university mathematics and beyond. It is also important to remember that algebraic manipulations might not necessarily find all solutions to a problem; you always need to reason carefully that all possibilities have been considered. Moreover, in complicated situations it is necessary to check that all proposed solutions unearthed by algebra are in fact valid solutions. Powers, roots and quadratics all link together very nicely when complex numbers are considered.

Quadratic equations and powers are commonly used throughout school and university mathematics and beyond. It is also important to remember that algebraic manipulations might not necessarily find all solutions to a problem; you always need to reason carefully that all possibilities have been considered. Moreover, in complicated situations it is necessary to check that all proposed solutions unearthed by algebra are in fact valid solutions. Powers, roots and quadratics all link together very nicely when complex numbers are considered.