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