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# Find all solutions to the system

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Find all solutions to the system

a + b = 10

a^3 + b^3 = 812

Oct 31, 2020

#1
+10583
+1

Find all solutions to the system
a + b = 10
a^3 + b^3 = 812

Hello Guest!

$$a + b = 10\\ a^3 + b^3 = 812$$

$$b=10-a\\ a^3+(10-a)(100-20a+a^2)=812\\ a^3+1000-200a+10a^2-100a+20a^2-a^3\\$$

$$30a^2-300a+1000=0\\ 3a^2-30a+100=0\\ a^2-10a+\frac{100}{3}=0\\ a=5\pm \sqrt{25-\frac{100}{3}}\\ a=5\pm i\sqrt{\frac{100}{3}-25}$$

$$a_1=5+i\cdot \frac{5\sqrt{3}}{3}\\ \color{blue}a_1=5+2.88675\ i\\ a_2=5-i\cdot \frac{5\sqrt{3}}{3}\\ \color{blue}a_2=5-2.88675\ i$$

$$b_1=10-(5+2.88675i)\\ \color{blue}b_1=5-2.88675\ i\\ b_2=10-(5-2.88675i)\\ \color{blue}b_1=5+2.88675\ i\\$$

!

Nov 2, 2020