Quadratic Formula Guide — Discriminant, Roots & Parabola

The quadratic formula

Any quadratic equation ax² + bx + c = 0 (with a ≠ 0) can be solved in one step using the quadratic formula.

The quadratic formula is: x = (−b ± √(b² − 4ac)) / (2a). The ± gives you the two roots. You plug in a, b, c and the formula returns both solutions directly.

Two things make it "one step": it works for every quadratic (factoring only works when the roots are nice), and it handles complex roots automatically when the discriminant is negative.

Derivation (for interest, not memorization): the formula comes from completing the square on ax² + bx + c = 0. The calculator does the arithmetic — your job is to identify a, b, c correctly from the equation.

What the discriminant (Δ = b² − 4ac) tells you

The sign of the discriminant classifies the roots before you compute them.

Δ > 0: two distinct real roots. The parabola crosses the x-axis at two points.

Δ = 0: one repeated real root (double root). The parabola touches the x-axis at exactly one point — the vertex.

Δ < 0: two complex conjugate roots of the form p ± qi. The parabola sits entirely above or below the x-axis and never crosses it.

In a physics or engineering problem, negative Δ usually means "no real solution" — e.g., a projectile never reaches a given height, or a circuit never oscillates at the requested frequency.

Vertex & the shape of the parabola

The vertex of y = ax² + bx + c sits at x = −b/(2a). Its y-value and the sign of a fully describe the parabola.

Vertex x-coordinate: always at −b/(2a). This is also the axis of symmetry — the parabola is a mirror image on either side.

Vertex y-coordinate: plug the vertex x back into f(x) = ax² + bx + c. That gives the minimum (if a > 0) or maximum (if a < 0) of the parabola.

Sign of a: positive a means the parabola opens upward (a smile shape). Negative a means it opens downward (a frown shape). The calculator marks the vertex and roots directly on the plot so you can see this instantly.

Real-world: the vertex is where motion reverses (a ball thrown up starts falling), where profit peaks before declining, or where a cable sags the most.

When does the factored form work?

The factored form (x − r₁)(x − r₂) only reads cleanly when the roots are rational numbers. For irrational or complex roots, the formula is your only route.

If Δ is a perfect square of a rational number (e.g., 1, 4, 9, 25, or 1/4), both roots are rational and the equation factors neatly over the integers or simple fractions.

If Δ is positive but not a perfect square (e.g., 5, 7, 12), the roots are irrational. You get answers like x = (3 ± √5) / 2 — still correct, just not factorable over the rationals.

If Δ is negative, the roots are complex. No real factored form exists; you need complex numbers to write (x − (p + qi))(x − (p − qi)).

Five sign-error traps to avoid

Most quadratic mistakes are sign errors in the discriminant. Work carefully with negative coefficients.

1. Losing a parenthesis on negative b. If b = −5, then b² = (−5)² = 25, not −25. Always parenthesize b when squaring.

2. Confusing −b with b. The formula starts with −b, so if b = −5, then −b = +5. Students often substitute b directly, forgetting the sign flip.

3. Sign error inside 4ac. If c is negative, 4ac is negative, and −4ac becomes positive. Double-check the sign of the final Δ.

4. Forgetting the ± gives two values. Always write out both roots: one with +, one with −. The answer isn&apos;t a single number.

5. Applying the formula to non-quadratics. If a = 0, the equation is linear — the formula divides by zero and breaks. The solver catches this and errors out.