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# help

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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