+26367 Find all solutions to the system
a + b = 14
a^3 + b^3 = 812 + 3ab
\(\begin{array}{|rcll|} \hline a+b &=& 14 \\ \mathbf{b} &=& \mathbf{14-a} \\ \hline (a+b)^3 &=& a^3+3a^2b+3ab^2+b^3 \\ (a+b)^3 &=& a^3+b^3 +3ab(a+b) \quad | \quad a^3 + b^3 = 812 + 3ab,~ a+b=14\\ 14^3 &=& 812 + 3ab+3*14*ab \\ \ldots \\ ab &=& \dfrac{1932}{45} \quad | \quad b = 14-a \\ a(14-a) &=& \dfrac{1932}{45}\\ 14a-a^2 &=& \dfrac{1932}{45}\\ \mathbf{a^2-14a+\dfrac{1932}{45} } &=& \mathbf{0} \\ a_1 &=& 9.4630604269 \\ a_2 &=& 4.5369395731\\\\ \qquad \mathbf{b} &=& \mathbf{14-a} \\ b_1 &=& x_2 \\ b_2 &=& x_1 \\ \hline \end{array}\)
+26367 Thank you Asinus:
Find all solutions to the system
a + b = 14
a^3 + b^3 = 812 + 3ab
\(\begin{array}{|rcll|} \hline a+b &=& 14 \\ \mathbf{b} &=& \mathbf{14-a} \\ \hline (a+b)^3 &=& a^3+3a^2b+3ab^2+b^3 \\ (a+b)^3 &=& a^3+b^3 +3ab(a+b) \quad | \quad a^3 + b^3 = 812 + 3ab,~ a+b=14\\ 14^3 &=& 812 + 3ab+3*14*ab \\ \ldots \\ ab &=& \dfrac{1932}{45} \quad | \quad b = 14-a \\ a(14-a) &=& \dfrac{1932}{45}\\ 14a-a^2 &=& \dfrac{1932}{45}\\ \mathbf{a^2-14a+\dfrac{1932}{45} } &=& \mathbf{0} \\ a_1 &=& 9.4630604269 \\ a_2 &=& 4.5369395731\\\\ \qquad \mathbf{b} &=& \mathbf{14-a} \\ b_1 &=& 14-a_1 \\ b_1 &=& a_2 \\ b_1 &=& 4.5369395731 \\\\ b_2 &=& 14-a_2 \\ b_2 &=& a_1 \\ b_2 &=& 9.4630604269 \\ \hline \end{array}\)
