Question

if x² + x =-1 , than what is the value of x³?

Answer 1

$\huge{\pink{\underline{Answer}}}$

x²+x=-1

Now,  x²+x+1=0
a=1
b=1
c=1  
$x=\frac{-b-\sqrt{b^{2}-4ac}}{2a}\\\implies \frac{-1-\sqrt{1-4}}{2}\\\implies \frac{-1-\sqrt{-3}}{2}\\\implies \frac{-1-1.732i}{2}$

$x=\frac{-b+\sqrt{b^{2}-4ac}}{2a}\\\implies \frac{-1+\sqrt{1-4}}{2}\\\implies \frac{-1+\sqrt{-3}}{2}\\\implies \frac{-1+1.732i}{2}$

Special solution

x²+x = -1

x²+x+1 = 0

multiply (x-1) on both side (x-1) is not equal to zero

(x-1)(x²+x+1) =0.(x-1)

Note :

(A) a³-b³=(a-b)(a²+ab+b²)

Now

x³-1³ = 0

x³ = 1³

substitute both sides

x³ = 1

Hence your value is 1