Find a + b = ? if a^3 + b^3 - 3ab =1 | Olympiad Question

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We rewrite the given condition in terms of a + b using the (a + b)^3 formula. Then use the product of all roots formulas to find the first solution for a + b. Then we use the completing the square method to find the second solution for a + b.

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I have a doubt. While getting product of roots, shouldn't it be (-1)(3ab - 1)/1.
Since 3ab is also a constant, and they together will form a°

ojaswinideshpande

Today I thought of another way if we take a+b=x and ab =y, then the equn becomes x^3__3xy+3y--1=0.regrouping, it is (x^3--1)--3y(x--1)=0 or (x--1)(x^2+x+1)--3y(x--1)=0 or (x--1)(x^2+x+1--3y)=0.so, x=1 and from the 2nd, treating it as a quadratic equation where a=1, b=1, c={1--3y) we get x= --1+-√{12y--3)/2. Here y=1 gives --1+--3whole divided by 2 or (3--1)/2=1 and (--1--3)/2=--2.so, x={a+b)=1, --2.

prabhudasmandal

Nice solution. For the first solution one has to point out that it can actually happen, for example, for a=1 and b=0.

laszloliptak

Sir i have another beautiful solution

a³+b³+(-1) ³= 3ab(-1)

So a+b+c= 0
Or a=b=c

So therefore a+b-1=0, a+b=1

a=b=-1 therefore a+b=-2

sohumsharma

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