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If a is not equal to 0, and if x + y = a and x^3 + y^3 = b, write an equation expressing x^2 + y^2 explicitly in terms of a and b.

 Dec 8, 2019
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x+y = a (1)

\(x^3+y^3=b\) (2)

 

\(x^3+y^3=(x+y)(x^2-xy+y^2)\) (4)

\((x+y)(x^2-xy+y^2)=b\) (3)

\(\frac{b}{(x+y)}=(x^2-xy+y^2)\)

From (1) we know that x+y=a

so

\(\frac{b}{a}=x^2-xy+y^2\) Expressed in terms of a and b as the question asks. and a can't be 0 as given. 

\(x^2+y^2=\frac{b}{a}+xy\) Expressed in terms of \(x^2+y^2\)

 

Notice that 

\((x+y)(x^2-xy+y^2)\)

we know \(x^2+y^2=\frac{b}{a}+xy\)

we can subsituite 

we also know (x+y)=a 

so

\(a*(\frac{b}{a}+xy-xy)\)

Simplify the brackets

\(a*(\frac{b}{a})\)

which is equal to b ,=x^3+y^3 as given.

So this identity is correct that x^2+y^2=b/a+xy

 Dec 8, 2019

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