Quadratic Formula Calculator

The Quadratic Formula

The quadratic formula solves any equation of the form ax² + bx + c = 0, where a ≠ 0. The formula is: x = (−b ± √(b² − 4ac)) / (2a). It was derived by completing the square on the general form and gives exact solutions for every quadratic equation, whether the roots are real, repeated, or complex. The formula is one of the most important in all of mathematics, appearing in physics, engineering, economics, and computer science.

Understanding the Discriminant

The expression under the square root, b² − 4ac, is called the discriminant and tells you everything about the nature of the roots before you solve. If the discriminant is positive, the parabola crosses the x-axis at two distinct points (two real roots). If it equals zero, the parabola just touches the x-axis (one repeated real root). If it is negative, the parabola never crosses the x-axis and the roots are complex (imaginary) numbers involving the imaginary unit i = √(−1).

The Vertex of a Parabola

Every quadratic equation ax² + bx + c = 0 corresponds to a parabola. The vertex — the highest or lowest point — is located at x = −b/(2a). Substituting back gives the y-coordinate: y = c − b²/(4a). The vertex is the axis of symmetry of the parabola and represents either the maximum (when a < 0) or minimum (when a > 0) value of the quadratic function.

Real-World Applications

Quadratic equations appear constantly in real life. Projectile motion — the path of a thrown ball or fired projectile — follows a quadratic equation where the discriminant tells you if and when it hits the ground. In business, profit functions that depend on price are often quadratic, with the vertex representing the profit-maximizing price. Engineers use quadratics in structural design, signal processing, and optics. The quadratic formula is not just an academic exercise — it is a practical tool for modeling the world.

FAQ

• What if a = 0? If a = 0, the equation is no longer quadratic — it becomes linear (bx + c = 0) and has one solution: x = −c/b.
• What are complex roots? When the discriminant is negative, the roots contain the imaginary unit i = √(−1). For example, x = 2 ± 3i means the roots are 2 + 3i and 2 − 3i. Complex roots always come in conjugate pairs.
• Can I factor instead of using the formula? Yes, when the roots are rational integers, factoring is faster. But the quadratic formula always works regardless of the roots' form.
• Why are there two roots? A parabola can cross the x-axis at most twice, so a quadratic can have up to two solutions. The ± in the formula generates both.