`x_1` and `x_2` are the roots of `a x^2+b x+c=0` and `x_1x_2<0.` Roots of `x_1(x-x_2)^2+x_2(x-x_1)^2=0` are: (a) real and of opposite sign b. negative c. positive d. none real

Options

A. real and opposite sign
B. negative
C. positive
D. nonreal

Answer

1

Explanation

`x_(1) (x - x_(2))^(2) + x_(2) (x - x_(1))^(2) = 0`
`rArr x^(2) (x_(1) + x_(2) ) - 4x a_(1) x_(2) + x_(1) x_(2) (x_(1) + x_(2)) = 0 `
` D = 16 (x_(1) x_(2))^(2) - 4x_(1) x_(2) (x_(1) + x_(2))^(2) gt (because x_(1) x_(2) lt 0 )`
The product of roots is `x_(1) x_(2) lt 0 ` . Then, the roots are real and
of opposite signs.