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Even-Type Quadratics

Problem

Suppose $x_1, x_2$ are the two real roots of the quadratic equation $x^2 + bx + c = 0$ , and $|x_1| + |x_2| = 2|k|$ for some integer $k$ . Then the equation is called an even-type quadratic. For example,

x^2 - 6x - 27 = 0,\quad x^2 - 2x - 8 = 0,\quad x^2 + 3x - \tfrac{27}{4} = 0,\quad x^2 + 6x - 27 = 0,\quad x^2 + 4x + 4 = 0

are all even-type quadratics.

1. Is $x^2 + x - 12 = 0$ an even-type quadratic? Justify your answer. 2. Let $b$ be an integer. Does there always exist a real number $c$ such that $x^2 + bx + c = 0$ is an even-type quadratic? Justify your answer.

Answer

Solution

Difficulty	5/10
Topics	Quadratic Equations, number theory, Absolute Value, algebra, Polynomials, Vieta's Formulas

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