Solve the equation 8x^6 + 7x^3 -1 = 0

The first thing to recognise is this is a quadratic in disguise, therefore we can rewrite the equation in terms of a new variable y.
Where y=x³
The equation then becomes 8y²+7y-1=0 .
We then factorise this into (8y-1)(y+1)=0 and work out y=1/8 or -1.

Then substitute this into the equation for y=x³ so that x³=1/8 and x³=-1
Solving for x gives us x=1/2 or -1
Things to note: A common mistake is that even though the square root of -1 has no solution the cube root of -1 does.