Determining When a Quadratic Expression is Negative: A Comprehensive Guide

Understanding when a quadratic expression is negative is a fundamental concept in algebra. In this article, we will guide you through the process of finding the values of p for which the quadratic expression (p^2 - 5p - 6) is negative. We will use a combination of analytical and graphical methods to provide a thorough explanation.

Factoring and Finding Roots

The first step in determining the negative interval for a quadratic expression is to factor it. The given expression is (p^2 - 5p - 6). Upon factoring, we obtain:

[p^2 - 5p - 6 (p - 3)(p 2)]

The roots of this expression are found by setting it equal to zero and solving for p:

[p - 3 0 quad text{or} quad p 2 0]

This results in the roots:

[p 3 quad text{or} quad p -2]

Sign Analysis and Negative Intervals

Once the roots are identified, we can use the sign analysis method to determine the intervals where the expression is negative. The roots divide the number line into three intervals: ((-∞, -2)), ((-2, 3)), and ((3, ∞)). We can test a point within each interval to determine the sign of the expression in that interval.

-2 p 3

Consider a test point within the interval ((-2, 3)), such as p 0:

[0^2 - 5(0) - 6 -6]

Since (-6

Tests Beyond the Roots

For intervals beyond the roots, we test points:

(p Consider p -3: