Solving cumbersome "quadratic" equation

Question

Solving the equation of the form

$$1-3x^2+3x\sqrt{1-2x^2}=0 $$

Is cumbersome since setting $t=1-2x^2$ does not yield an explicit quadratic formula in terms of t. There is some trix to this, but I already tried to solve it looking at $\sqrt{1-2x^2} $ as the $b$ in $ax^2+bx+c=0$ and also to insert $t$ as above and that yields nothing good. How does one go around this?

Answer 1

HINT:

We have $$3x\sqrt{1-2x^2}=3x^2-1$$

Square both sides to form a Quadratic Equation in $x^2$

Solve it & identify the extraneous roots introduced due to squaring

Answer 2

Hint:

Another way is to set $$ \sin \theta = \sqrt 2 \ x$$ and then solve an equation of the form:

$$ a\cos \phi + b \sin \phi = c $$