The equations x^3 + 5x^2 + px + q = 0 and x^3 - 5x^2 + px + r = 0 have two roots in common.
If the third root of each equation is represented by x_1 and x_2 respectively,
compute the ordered pair (x_1, x_2).
+397 Suppose that the common roots are a and b and that the other, individual roots, are m and n.
We then have (Veita),
a + b + m = -5, .................(1)
a + b + n = 5, .................(2)
ab + bm + ma = p, ...........(3)
ab + bn + na = p, .............(4).
So,
(2) - (1): n - m = 10, ........(5)
(4) - (3): b(n - m) + a(n - m) = 0, .....(6).
from which
b + a = 0
and then, from (1) and (2),
m = -5 and n = 5.
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