If (cosp-1) x^2+x cosp+sinp=0 has real roots, the interval...

If $(\cos p-1) x^{2}+x \cos p+\sin p=0$ has real roots, the interval of possible values of $p$ is
(a) $(-\pi, 0)$
(b) $(0, \pi)$
(c) $\left(\frac{-\pi}{2}, \frac{\pi}{2}\right)$
(d) $\left(0, \frac{3 \pi}{2}\right)$

Key Concepts

Quadratic Equations

A quadratic equation in the form ax² + bx + c = 0 is fundamental in algebra. Its solutions depend on the coefficients and can be found using the quadratic formula, completing the square, or factoring when possible. Understanding the structure of quadratic equations is crucial in many areas of mathematics, including when these coefficients depend on parameters.

Discriminant Condition

For a quadratic equation to have real roots, the discriminant, defined as b² - 4ac, must be nonnegative. This condition provides a direct way to determine if the equation yields real solutions, and it becomes particularly important when the coefficients are expressed in terms of another variable or parameter.

Parametric Coefficients

When the coefficients of a quadratic equation are functions of a parameter, analysis involves studying how changes in the parameter affect the nature of the roots. This often requires setting up and solving inequalities involving the discriminant to find the parameter values that guarantee real roots.

Trigonometric Functions

Trigonometric functions such as sine and cosine have well-known properties and ranges. When these functions appear in the coefficients of an equation, it is important to use their periodic and bounded nature to analyze how they influence the equation. Their behavior often provides additional constraints when solving problems that involve parameters.

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