√(x^2-5 x+6) ≤x+4

Name: Problem 245, Chapter 7: Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry
Uploaded: 2022-01-13T14:54:09-08:00
Duration: 1 min 18 s
Channel: Aman Gupta
Description: √(x^2-5 x+6) ≤x+4

Key Concepts

Domain of Square Root Functions

When working with square roots in an inequality, a key concept is determining the domain of the function, which means ensuring that the expression under the square root is nonnegative. This is essential because the square root of a negative number is not defined within the real numbers, and thus any solution must satisfy the underlying condition that the radicand is greater than or equal to zero.

Quadratic Factoring and Sign Analysis

The expression under the square root is a quadratic, and factoring or using the quadratic formula can help to identify its roots. This enables a sign analysis of the quadratic to determine the intervals where it is nonnegative. Understanding how the sign of a quadratic expression changes across its roots is critical when establishing the domain for the radical.

Inequality Solving Techniques

Solving inequalities that involve radicals often requires isolating the radical expression before proceeding. Once isolated, a common technique involves squaring both sides to eliminate the square root. However, caution must be exercised because this step can potentially introduce extraneous solutions. The overall process involves careful handling of the inequality while respecting all domain restrictions.

Extraneous Solutions

When squaring both sides of an equation or inequality, extra solutions that satisfy the squared form but not the original condition might appear. Verifying each candidate solution against the original inequality is crucial to ensure that only valid answers are included. This process of checking underscores the importance of considering the initial constraints imposed by the domain of the square root.

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