Q.199. Roots of a polynomial equation.

Question 199.
For real number x and y, if x^2 + xy - y^2 + 2x - y + 1 = 0, then which of the following is true?
A) y can't be between 0 and 8/5; B) y can't be between -8/5 and 8/5;
C) y can't be between -8/5 and 0; D) none.

Answer 199.
x^2 + xy - y^2 + 2x - y + 1 = 0
=> x^2 + (y + 2)x - (y^2 + y - 1) = 0
For real x,
discriminant of this quadratic equation in x ≥ 0
=> (y + 2)^2 + 4(y^2 + y - 1)  ≥ 0
=> 5y^2 + 8y ≥ 0
=> y(5y + 8) ≥ 0
 => y ≥ 0 and 5y + 8 ≥ 0 => y ≥ - 8/5
=> y ≥ 0
or
=> y ≤ 0 and 5y + 8 ≤ 0 => y ≤ - 8/5
=> y ≤ - 8/5
 => C) y can't be between - 8/5 and 0.

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