Equations `x^3+5x 62+p x+q=0a n dx 63+7x^2+p x+r=0` have two roots in common. If the third root of each equation is `x_1a n dx_2` , respectively, then find the ordered pair `(x_1, x_2^)dot`

Explanation

Let roots of `x^(3)+5X^(2) + px + q = 0 be alpha, beta, x_(1)`. Then
`alpha + beta + x_(1) = -5`
`alpha beta + betax_(1) + alphax_(1) = p` (1)
Roots of `x^(3) + 7x^(2) + px +r = 0 ` will be `alpha, beta, x_(2)`. Then
`alpha + beta + x_(2) = -7`
`alpha beta + betax_(2) + alphax_(2) = p` (2)
Subtracting (2) from (1), we get
`(x_(1) -x_(2)) (alpha + beta) = 0 (because x_(1) ne x_(2))`
`therefore alpha + beta = 0`
`rArr x_(1) = -5 and x_(2) = -7`