Let α, βbe real numbers and z a complex number. If z^2+αz+β=0 has two distinct roots on the line

step 1 The equation we are given an equation which states that z square plus alpha z plus beta is equal

Let α, βbe real numbers and z a complex number. If...

Let $\alpha, \beta$ be real numbers and $z$ a complex number. If $z^{2}+\alpha z+\beta=0$ has two distinct roots on the line $\operatorname{Re}(z)=1$, then it is necessary that
(A) $\beta \in(-1,0)$
(B) $|\beta|=1$
(C) $\beta \in(1, \infty)$
(D) $\beta \in(0,1)$

Key Concepts

Geometric Representation of Complex Numbers

Complex numbers can be represented as points in the complex plane, with the horizontal axis representing the real part and the vertical axis representing the imaginary part. Specifying that roots lie on a particular line, such as Re(z)=1, establishes a geometric constraint which can be used to characterize the algebraic properties of the quadratic equation, as the fixed real part influences the formation of the roots.

Quadratic Equations with Complex Roots

A quadratic equation with real coefficients can have either two real roots or a pair of complex conjugate roots. When the discriminant is negative, the roots are complex and occur in conjugate pairs. Understanding the nature of these roots is fundamental in solving quadratic equations and analyzing their behavior in the complex plane.

Vieta's Formulas

Vieta's formulas relate the sum and product of the roots of a polynomial to its coefficients. For a quadratic equation, the sum of the roots equals -? and the product equals ?. When the roots are expressed in a form that reflects a geometric constraint, such as having a fixed real part, these relationships help in deducing necessary conditions on the coefficients.

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