Quadratic Formula

Frequently Asked Questions

What is the quadratic formula?

For ax² + bx + c = 0, x = (−b ± √(b² − 4ac)) / (2a). It always works as long as a ≠ 0.

What does the discriminant tell me about the roots?

Discriminant b² − 4ac determines root types: positive → two distinct real roots, zero → one repeated real root, negative → two complex conjugate roots.

When should I factor instead of using the formula?

Factor when the equation has small integer roots - like x² − 5x + 6 = (x − 2)(x − 3). Use the formula when factoring is not obvious.

Where do quadratics appear in real life?

Projectile motion, parabolic antennas, profit maximization, and the trajectory of anything thrown under gravity all follow quadratic equations.

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How This Calculator Works

Enter the three coefficients a, b, and c of a quadratic equation in the form ax² + bx + c = 0. The calculator evaluates the discriminant Δ = b² − 4ac, then applies the quadratic formula x = (−b ± √Δ) / (2a) to find the roots. When Δ is negative, complex roots are returned in the form p ± qi.

The Discriminant Explained

Δ > 0 (two distinct real roots): The parabola crosses the x-axis at two different points. Example: x² − 5x + 6 = 0 has roots x = 2 and x = 3.

Δ = 0 (one repeated real root): The parabola’s vertex sits exactly on the x-axis. The equation has one solution: x = −b / (2a).

Δ < 0 (two complex roots): The parabola does not intersect the x-axis at all. The roots involve the imaginary unit i = √−1 and always come as a conjugate pair: p + qi and p − qi.

Methods for Solving Quadratics

Factoring: Fastest when integer roots exist. Rewrite ax² + bx + c as a product of two linear factors. Not always possible with nice numbers.

Completing the square: Rewrites the equation in vertex form a(x − h)² + k = 0. Useful for identifying the vertex and deriving the quadratic formula itself.

Quadratic formula: Always works, for any real values of a, b, c (a ≠ 0). The universal method of last resort - and first choice when factoring is not obvious.

Parabola Properties

The graph of y = ax² + bx + c is a parabola. Its axis of symmetry is the vertical line x = −b / (2a), and the vertex sits at that x-coordinate. If a > 0 the parabola opens upward (minimum at vertex); if a < 0 it opens downward (maximum at vertex). The vertex form y = a(x − h)² + k makes these properties immediately visible.

Real-World Applications

Projectile motion: The height h(t) = −4.9t² + v₀t + h₀ of a thrown object is quadratic in time. Setting h = 0 and solving gives the landing time.

Profit maximization: Revenue minus cost often yields a quadratic in price or quantity; the vertex gives the optimal price.

Area problems: Finding dimensions of a rectangle given a fixed perimeter and area leads directly to a quadratic equation.

Quadratic Formula

Frequently Asked Questions

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