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Wednesday, June 9, 2021

2021/042) For real numbers a and b that satisfy a^3+12a^2+49a + 69=0 and b^3 -9b^2+ 28b -31 = 0 find a+b.

We have
 a^3+12a^2+49a + 69=0
=> (a+4)^3 + (a+5) = 0
or (a+4)^3 + (a+4) + 1= 0\cdots(1)

b^3 -9b^2+ 28b -31 = 0
=>(b-3)^3 + (b-4) = 0
=>(b-3)^3 + (b-3) -1 = 0\cdots(2)

if we define f(x) = x^3+x knowing that f(x) is odd function
(f(a+4) = -1 and -f(b-3) = 1

so (a+4) = - (b- 3)

or a + b = - 1

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