Enter a, b, and c

Type the three coefficients a, b, and c into the labeled inputs (or tap a preset for a quick example — note that a must not be zero).

2

Read the Results

Review the discriminant badge for root type, both simplified roots, and vertex/axis-of-symmetry/y-intercept sub-stats.

3

View the Parabola

Inspect the labeled parabola chart for roots, vertex, and axis of symmetry, then toggle "Show Step-by-Step Work" for the 5-step solution.

4

Learn with the Lesson Tab

Walk through the personalized derivation on the Lesson tab, then visit Parabola Explorer to see how each coefficient reshapes the curve.

Quadratic Formula

x = (−b ± √(b²−4ac)) / (2a)

Solves ax² + bx + c = 0 for x; a, b, c are the polynomial coefficients. The ± produces up to two solutions for the equation's roots.

Discriminant

Δ = b² − 4ac

The value under the square root. Positive Δ gives two real roots, Δ = 0 gives one repeated root, negative Δ gives two complex conjugate roots.

Vertex Coordinates

Vertex (h, k): h = −b / (2a); k = a·h² + b·h + c

The vertex is the parabola's turning point — minimum when a > 0 (opens up) or maximum when a < 0 (opens down). The axis of symmetry is x = h.

Discriminant The value b²−4ac that determines the number and type of roots. Positive → two real roots, zero → one repeated root, negative → two complex roots.

Roots / Zeros The x values that satisfy ax²+bx+c=0. Geometrically, they are the x-intercepts of the parabola (when real).

Vertex The turning point of the parabola. It is the minimum when a>0 (opens up) and the maximum when a<0 (opens down).

Axis of Symmetry The vertical line x = −b/(2a) that passes through the vertex and splits the parabola into two mirror images.

Parabola The U-shaped curve described by a quadratic function. It is symmetric about its axis of symmetry and has one vertex.

Complex Number A number of the form a + bi where i = √(−1). Complex roots always appear in conjugate pairs (a+bi and a−bi) for equations with real coefficients.

x²

x² − 5x + 6 = 0

Two Distinct Real Roots

Discriminant Δ = 25 − 24 = 1

x₁ = (5 + 1) / 2 = 3 and x₂ = (5 − 1) / 2 = 2. Vertex at (2.5, −0.25). The parabola crosses the x-axis at x = 2 and x = 3.

x²

x² − 2x + 1 = 0

One Repeated Root

Discriminant Δ = 4 − 4 = 0

x = 2 / 2 = 1 (one repeated root). This is a perfect square trinomial: (x−1)² = 0. The parabola touches the x-axis at exactly one point: its vertex (1, 0).

x²

x² + x + 1 = 0

Two Complex Roots

Discriminant Δ = 1 − 4 = −3

x = −1/2 ± (√3/2)i. Since Δ < 0 the parabola has no real x-intercepts — it floats entirely above the x-axis. The complex conjugate roots are ≈ −0.5 ± 0.866i.

Quadratic equations (polynomials of the form ax² + bx + c = 0) appear throughout physics, engineering, economics, and applied mathematics because they describe the behavior of systems with constant acceleration, parabolic trajectories, maximized or minimized quantities, and inverse-square relationships. The quadratic formula solves them all in closed form, which is why it's one of the most memorized equations in high school math curricula worldwide. The sections below cover three real-world domains where quadratic equations drive practical decisions — projectile motion in physics, profit optimization in business, and structural and electrical engineering applications — showing why memorizing the formula is less valuable than understanding when and why to reach for it.

Projectile Motion

When you throw a ball upward, its height h(t) at time t follows the quadratic equation h(t) = −½gt² + v₀t + h₀, where g is gravitational acceleration (9.81 m/s² on Earth), v₀ is initial upward velocity, and h₀ is starting height. Solving h(t) = 0 with the quadratic formula gives the times when the ball hits the ground — often two solutions where the negative one is physically meaningless (time before launch) and the positive one is the actual flight duration.

Engineers use this equation in ballistics (artillery firing tables, sniper calculations), sports science (optimal basketball arc trajectories, maximum javelin range angles, baseball pitch analysis), rocket trajectory planning, and drone flight control. The vertex (h(t_v), t_v where t_v = v₀/g) gives maximum height reached, useful for clearance-over-obstacle calculations. For anything beyond simple vertical motion, the quadratic form extends to full 2D projectile motion with horizontal and vertical components, and the same quadratic formula applies to solving for flight time, range, and apex height. Air resistance complicates real-world projectiles, but the ideal quadratic model is accurate enough for most practical calculations in Earth's atmosphere at low speeds.

Business Profit Optimization

Revenue and cost functions in economics are often quadratic because they involve pricing decisions that interact with demand in ways that produce parabolic profit curves. For example, if a company sets price p and the demand function is Q = 100 − 2p (demand decreases linearly as price rises), then revenue R = p·Q = 100p − 2p² — a downward-opening parabola. To find the profit-maximizing price, you take the derivative and set it to zero (dR/dp = 100 − 4p = 0, giving p = 25), which is equivalent to finding the vertex of the parabola.

The quadratic formula is also used to find break-even points where profit = 0: set revenue minus costs equal to zero and solve for the production quantity or price that makes the business break even. A company with fixed costs $500, variable cost $10/unit, and selling price $50/unit has profit = 40Q − 500 (linear, not quadratic), but with volume-dependent pricing (p = 50 − 0.1Q), profit becomes −0.1Q² + 40Q − 500, and the quadratic formula gives break-even at Q ≈ 13 and Q ≈ 387, with maximum profit at Q = 200. This kind of analysis drives pricing strategy, production planning, and the "what volume makes this product worth making?" decisions in manufacturing and services.

Physics and Engineering

Quadratic equations appear throughout physics and engineering in forms that require quick and accurate solutions. Electrical power dissipation (P = I²R) creates a quadratic relationship between current and power, used in sizing conductors and heat-sink requirements. Lens formulas in optics relate focal length, object distance, and image distance in quadratic forms that solve for image positions in camera and telescope design. The equations of motion for objects under constant force are all quadratic in time, which is why projectile and freefall problems solve cleanly with the quadratic formula.

In structural engineering, the shape of arches and suspension cables under uniform loading follows parabolic equations, and load calculations require solving quadratic inequalities to ensure safety margins are met (stress must stay below material yield limits across the span). Beam deflection under distributed loads produces fourth-order differential equations whose solutions involve nested quadratic terms. Civil engineers designing bridges, buildings, and dams use quadratic analysis routinely for wind-load calculations, column-buckling analysis, and foundation-settlement predictions. The quadratic formula's combination of closed-form solvability and widespread applicability makes it one of the highest-leverage equations in any engineering curriculum.

What is the quadratic formula?+

x = (−b ± √(b²−4ac)) / (2a). It solves any equation of the form ax²+bx+c=0 for the values of x. The ± means there are two solutions: one with + and one with −.

What is the discriminant and what does it tell me?+

Δ = b²−4ac. If Δ > 0 there are two distinct real roots (the parabola crosses the x-axis twice). If Δ = 0 there is one repeated root (the parabola just touches the x-axis). If Δ < 0 there are two complex roots (no real x-intercepts).

What are complex roots and how do I interpret them?+

Complex roots take the form a ± bi where i = √(−1). They mean the parabola never crosses the x-axis — it is entirely above (if a > 0) or below (if a < 0) the x-axis. Complex roots always appear in conjugate pairs for equations with real coefficients.

How is the vertex calculated?+

The vertex x-coordinate is h = −b/(2a). Substitute back: k = a·h²+b·h+c. This gives the turning point (h, k). When a > 0 the vertex is the minimum; when a < 0 it's the maximum.

When should I use the quadratic formula vs factoring?+

Factoring is faster when integer roots exist (discriminant is a perfect square, small coefficients). Use the quadratic formula for decimal coefficients, large numbers, or when you need complex roots. The formula always works regardless of root type.

Why does a have to be non-zero?+

If a = 0 the equation becomes bx+c = 0 — a linear equation, not quadratic. The term ax² disappears and the quadratic formula involves dividing by 2a = 0, which is undefined. Linear equations have at most one solution and are solved differently.

What is vertex form and how do I convert to it?+

Vertex form is a(x−h)²+k. Compute h = −b/(2a) and k = a·h²+b·h+c, then substitute. For example, x²−4x+3 → h = 2, k = −1, so vertex form is (x−2)²−1.

What does the chart show?+

The parabola chart marks real roots as cyan dots on the x-axis, the vertex as a gold dot, and the y-intercept as a teal dot. The dashed vertical line is the axis of symmetry at x = −b/(2a). For equations with complex roots the parabola does not touch the x-axis.

Can I solve equations with non-integer coefficients?+

Yes. Enter decimals directly (e.g. a = 1.5, b = −2.7, c = 0.3). The calculator handles any real-number coefficients and displays roots as simplified fractions when possible, or as decimals otherwise.

What is the axis of symmetry?+

The vertical line x = −b/(2a) that splits the parabola into two mirror images. Every parabola has exactly one axis of symmetry, and it always passes through the vertex.

Quadratic Formula Calculator

Step-by-Step Solution

Identify a, b, and c

Calculate the discriminant Δ = b²−4ac

Apply + case: (−b + √Δ) / 2a

Apply − case: (−b − √Δ) / 2a

Final Roots

Adjust Coefficients

Coefficient a

Coefficient b

Coefficient c

How to Use This Calculator

Enter a, b, and c

Read the Results

View the Parabola

Learn with the Lesson Tab

Formula & Methodology

Key Terms Explained

Real-World Examples

x² − 5x + 6 = 0

x² − 2x + 1 = 0

x² + x + 1 = 0

Quadratic Equations in the Real World

Projectile Motion

Business Profit Optimization

Physics and Engineering

Frequently Asked Questions

Step-by-Step Solution

Identify a, b, and c

Calculate the discriminant Δ = b²−4ac

Apply + case: (−b + √Δ) / 2a

Apply − case: (−b − √Δ) / 2a

Final Roots

Adjust Coefficients

Coefficient a

Coefficient b

Coefficient c

How to Use This Calculator

Enter a, b, and c

Read the Results

View the Parabola

Learn with the Lesson Tab

Formula & Methodology

Key Terms Explained

Real-World Examples

Quadratic Equations in the Real World

Projectile Motion

Business Profit Optimization

Physics and Engineering

Frequently Asked Questions

Keep Exploring

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