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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).

 Oct 28, 2021
 #1
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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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 Oct 29, 2021

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