Prove Existence Unique Real Solution

[knowLittle]

Homework Statement

Prove Existence Unique Real Solution to
$x^{3} + x^{2} -1 =0$ between $x= \frac{2}{3} \text{and} x=1$

The Attempt at a Solution

$x^{2} ( x+1) =1$
I know that the solution is x =0.75488, but this came from some website. How do I find this number without a calculator?

 

[micromass]

You don't actually need to find the solution. You just need to know it exists. Think intermediate value theorem.

 

[HallsofIvy]

The intermediate value theorem shows that there exist a solution betweem 2/3 and 1. The mean value theorem shows that there is only one.