A plus b equals 1. A squared plus b squared is 2. What is a cubed plus b cubed

A + b = 1  →  b = 1 - A 

So, subsituting, we have

A² + (1 - A)² = 2

A² + 1 - 2A  + A² = 2    rearrange

2a² - 2A - 1 = 0      And using the on-site solver and substituting "x" for "A" ......we have....

$${\mathtt{2}}{\mathtt{\,\times\,}}{{\mathtt{x}}}^{{\mathtt{2}}}{\mathtt{\,-\,}}{\mathtt{2}}{\mathtt{\,\times\,}}{\mathtt{x}}{\mathtt{\,-\,}}{\mathtt{1}} = {\mathtt{0}} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = {\mathtt{\,-\,}}{\frac{\left({\sqrt{{\mathtt{3}}}}{\mathtt{\,-\,}}{\mathtt{1}}\right)}{{\mathtt{2}}}}\\
{\mathtt{x}} = {\frac{\left({\sqrt{{\mathtt{3}}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{1}}\right)}{{\mathtt{2}}}}\\
\end{array} \right\} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = -{\mathtt{0.366\: \!025\: \!403\: \!784\: \!438\: \!6}}\\
{\mathtt{x}} = {\mathtt{1.366\: \!025\: \!403\: \!784\: \!438\: \!6}}\\
\end{array} \right\}$$   

So, b = 1 - (-[√3 - 1 ] /  2)   =  [1 + √3 ] / 2     or  b = 1 - [ [1 + √3 ] / 2] = [1 - √3 ] / 2

So

A³ + b³ = ( [1 - √3] / 2 )³  + ( [1 + √3 ] / 2)³ = ( [1 + √3] / 2 )³  + ( [1 - √3 ] / 2)³ = 2.5

A plus b equals 1. A squared plus b squared is 2. What is a cubed plus b cubed

$$a+b=1 \\
a^2+b^2 = 2\\
a^3+b^3 = ?$$

I.

$$a^3+b^3 = (a+b)(a^2-ab+b^2) \quad | \quad a+b=1 \ and \ a^2 +b^2 = 2 \\
a^3+b^3 = 1*(2-ab) \\
a^3 + b^3 = 2 -ab$$

II.

$$(a+b)^2 = a^2+ 2ab + b^2 \quad | \quad a+b=1 \ and \ a^2+b^2 = 2\\
1^2 = 2 + 2ab \\
-1 = 2ab \\
ab = -\frac{1}{2}$$

III.

$$a^3+b^3 = 2-ab \quad | \quad ab = -\frac{1}{2} \\
a^3+b^3 = 2 - ( -\frac{1}{2} ) \\
a^3+b^3 = 2 + \frac{1}{2} \\
\boxed{a^3+b^3 = 2.5}$$