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.

Guest Dec 8, 2019

#1**0 **

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

Guest Dec 8, 2019