Quadratic Formula and Discriminant

Objective: Use standard-form coefficients in the quadratic formula.

Prerequisite skills: standard-form quadratics and square roots. For ax² + bx + c = 0, identify a, b, and c with their signs. Substitute into the formula and evaluate the discriminant before the square root. Keep the entire numerator over 2a. Common error: dividing only the square-root term by 2a. Restrictions and interpretation: Use a ≠ 0; the discriminant classifies real roots, and a repeated root counts once.

Objective: Use the discriminant to count real roots.

The quantity D = b² − 4ac determines the real-root count: two for D > 0, one for D = 0, none for D < 0. A positive nonsquare discriminant still gives two real roots. Common error: assuming a non-square discriminant means no real roots. Restrictions and interpretation: Use a ≠ 0; the discriminant classifies real roots, and a repeated root counts once.

Objective: Set the discriminant equal to zero to find a tangency condition.

Exactly one distinct real solution occurs at D = 0. Solve that equation for the unknown parameter, then verify the quadratic coefficient remains nonzero. The condition separates the two-root and no-real-root cases. Common error: setting the discriminant positive when one distinct root is required. Restrictions and interpretation: Use a ≠ 0; the discriminant classifies real roots, and a repeated root counts once. 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.