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If x^2 + y^2 = 7 and x^3 + y^3 = 10, then find the value of x + y.

Jun 19, 2020

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We express x^2 + y^2 and x^3 + y^3 in terms of (x + y) and (xy).

$$x^2 + y^2 = (x + y)^2 - 2(xy)\\ x^3 + y^3 = (x + y)((x + y)^2 - 3(xy))$$

Now, let x + y = t, xy = u.

$$\begin{cases}t^2 - 2u = 7\\t(t^2 - 3u)=10\end{cases}$$

Substituting the first equation into the second,

$$\begin{cases}t^2 - 2u = 7\\t(7 - u)=10\end{cases}$$

Manipulating the second equation, $$u = 7 - \dfrac{10}t$$

Substituting,

$$t^2 - 14 + \dfrac{20}t = 7\\ t^3 - 21t + 20 = 0$$

By factor theorem, (t - 1) is a factor of the left hand side.

$$(t - 1)(t^2 + t - 20) = 0\\ (t - 1)(t - 4)(t + 5) = 0\\ t = 1 \text{ or }t = 4 \text{ or } t= -5$$

Therefore, the possible values of (x + y) are 1, 4, and -5.

Jun 19, 2020