find x, y ---Question Stem:
find x, y that $(3+x\sqrt{5})(\sqrt{5}-y) = -13+5\sqrt{5}$
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We need to find x and y that satisfy the equation (3 + x√5)(√5 - y) = -13 + 5√5. Let's start by expanding the left side of the equation. When we multiply out the terms, we get 3√5 - 3y + 5x - xy√5. We can group the rational and irrational terms to get (5x - 3y) + (3 - xy)√5.
Now we equate the rational and irrational parts separately. The expanded left side (5x - 3y) + (3 - xy)√5 must equal -13 + 5√5. This gives us two equations: the rational parts give us 5x - 3y = -13, and the irrational parts give us 3 - xy = 5. Simplifying the second equation, we get xy = -2.
Now we solve the system of equations. From xy = -2, we can express y as -2/x. Substituting this into the first equation 5x - 3y = -13, we get 5x + 6/x = -13. Multiplying through by x gives us 5x² + 6 = -13x. Rearranging, we obtain the quadratic equation 5x² + 13x + 6 = 0.
Now we factor the quadratic equation 5x² + 13x + 6 = 0. We need two numbers that multiply to 30 and add to 13, which are 10 and 3. We rewrite as 5x² + 10x + 3x + 6 = 0, then group: 5x(x + 2) + 3(x + 2) = 0. This factors to (x + 2)(5x + 3) = 0, giving us x = -2 or x = -3/5.
Finally, we find the corresponding y values using y = -2/x. For x = -2, we get y = 1, giving us the solution (-2, 1). For x = -3/5, we get y = 10/3, giving us the solution (-3/5, 10/3). Therefore, the pairs (x, y) that satisfy the original equation are (-2, 1) and (-3/5, 10/3).