Challenging Problems with Unknown Constants and Constraints

Objective: Use a specified point to recover a model constant.

Prerequisite skills: systems, quadratic discriminants, and domain restrictions. An unknown constant becomes constrained when a graph passes through a given point or an equation has a stated solution. Substitute the complete point and solve for the parameter. If several conditions are given, test them all. Common error: using only one coordinate of a specified point. Restrictions and interpretation: Apply every stated condition: nonzero coefficients, real solutions, positivity, distinctness, and domain exclusions can change a parameter result.

Objective: Identify a sharp threshold for two real intersections.

A parameter can change a discriminant continuously. Solve the appropriate strict or non-strict discriminant inequality, and distinguish the boundary case from the interval satisfying the required count. Do not treat “two” as “at least one.” Common error: including a repeated-root boundary in a two-distinct-root condition. Restrictions and interpretation: Apply every stated condition: nonzero coefficients, real solutions, positivity, distinctness, and domain exclusions can change a parameter result.

Objective: Combine circle geometry and tangency in a parameter condition.

For a centered circle and a horizontal line, tangency occurs when the vertical center-to-line distance equals the radius. Translate that geometry into the circle parameter and verify a positive radius. This uses an exact geometric condition instead of an approximate graph touch. Common error: equating the radius squared to an unsquared distance. Restrictions and interpretation: Apply every stated condition: nonzero coefficients, real solutions, positivity, distinctness, and domain exclusions can change a parameter result. Desmos and method choice: Use Desmos for arithmetic or a graph/table check when that reduces work. Enter parentheses explicitly and verify the requested quantity and units. Use the algebraic steps shown here for an exact result; a rounded graph display is not proof of exact equality.