If c<a<b<d, then roots of the equation b x^2+(1-b (c+d)) x+b...

If $c<a<b<d$, then roots of the equation $b x^{2}+(1-b$ $(c+d)) x+b c d-a=0$
(A) are real and one lies between $c$ and $a$
(B) real and distinct in which one lies between $a$ and $b$
(C) real and distinct in which one lies between $c$ and $d$
(D) roots are not real

Key Concepts

Vieta's Formulas

Vieta's formulas relate the coefficients of a quadratic equation to the sum and product of its roots. Specifically, for an equation of the form ax² + bx + c = 0, the sum of the roots is -b/a and the product is c/a. These relationships are particularly useful in analyzing and drawing conclusions about the roots without explicitly computing them.

Intermediate Value Theorem and Interval Analysis

The Intermediate Value Theorem states that if a continuous function takes on values of opposite signs at two points, it must cross zero between those points, indicating the presence of a root. Combined with sign analysis of the polynomial, this theorem helps determine the specific intervals in which the real roots lie, a technique that is crucial for understanding the location and behavior of roots within given bounds.

Quadratic Equations

A quadratic equation is a second-degree polynomial equation in one variable, typically expressed in the form ax² + bx + c = 0. Understanding its structure and the methods used to solve it—such as factoring, completing the square, or using the quadratic formula—is essential for analyzing the behavior of its roots.

Quadratic Discriminant

The discriminant of a quadratic equation, given by b² - 4ac, determines the nature of the roots. A positive discriminant indicates that the equation has two real and distinct roots, a zero discriminant indicates a single repeated real root, and a negative discriminant implies the roots are complex. This concept is fundamental when assessing the reality of the roots.

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