Prove that there exists a unique real number solution to the equation x3 + x2 − 1 = 0 between x = 2/3 and x=1

Replace each of the x values given to see if they give 0. 

0=(2/3)³+(2/3)²-1 
0=(8/27)+(4/9)-1 
0≠-0.259

0=(1)³+(1)²-1 
0≠1

Therefore, you can see that the solution would have to be between 2/3 and 1.

Hope I helped :)