Unknown Coefficients and Conditions on Solutions

Objective: Find a coefficient from a stated root.

Prerequisite skills: discriminants, factored form, and quadratic roots. A root r satisfies f(r) = 0. Substitute that value into the polynomial and solve the resulting coefficient equation. A root condition alone does not generally imply that the root is repeated. Common error: setting every coefficient to zero when only one root is specified. Restrictions and interpretation: Check a ≠ 0 before applying quadratic conditions; distinguish a specified root from a repeated root.

Objective: Relate unknown coefficients to the roots.

For ax² + bx + c with a ≠ 0 and roots r,s, expansion of a(x−r)(x−s) gives r+s=−b/a and rs=c/a. These relations can answer coefficient questions without finding individual roots. Common error: forgetting the negative sign in the root sum. Restrictions and interpretation: Check a ≠ 0 before applying quadratic conditions; distinguish a specified root from a repeated root.

Objective: Solve a discriminant inequality for a parameter.

At least one real quadratic solution requires D ≥ 0; two distinct real solutions require D > 0. An equality case must be included only when a repeated root is allowed. Common error: excluding equality when at least one real root is requested. Restrictions and interpretation: Check a ≠ 0 before applying quadratic conditions; distinguish a specified root from a repeated root. 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.