Question 20

If $$x^3 + y^3 = 9$$ and $$x + y = 3$$, then find the value of $$x^2 + y^2$$

Solution

$$ x^3 + y^3 = 9 $$

x + y = 3

$$ (x + y)^3 = 3^3 $$

$$ x^3 + y^3 + 3xy(x + y) = 27 $$

9 + 3xy(x + y) = 27

3xy(x + y) = 18

xy(x + y) = 6

$$ xy \times 3 = 6 $$

xy = 2

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

$$ x^2 + y^2 = 3^2 - 2 \times 2 $$

$$ x^2 + y^2 = 9 - 4 = 5 $$

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