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Quadratic Equation Solver

Math

1x² − 3x + 2 = 0

Coefficients

Must be ≠ 0
Two Distinct Real Roots

Discriminant (D)

1

b² − 4ac

Root 1 (x₁)

2

Root 2 (x₂)

1

Parabola Properties

Vertex

(1.5, -0.25)

Axis of Symmetry

x = 1.5

Y-Intercept

2

Opening

↑ Upward

Vertex Form

(x − 1.5)² − 0.25

Factored Form

(x − 2)(x − 1)

Parabola Graph

Parabola

Axis of symmetry

Vertex

Root(s)

About This Tool

🧮 Quadratic Equation Solver – Roots, Vertex & Parabola Properties

A quadratic equation is any polynomial of degree two written in the standard form ax² + bx + c = 0, where a ≠ 0. Quadratics appear everywhere — from the arc of a thrown ball to the profit curves of a business, from the lens curvature in optics to the signal processing algorithms in your phone. This solver computes every meaningful property of a quadratic equation in one place: roots, discriminant, vertex, axis of symmetry, factored form, and vertex form, along with an interactive parabola graph.

📐 The Quadratic Formula

The cornerstone of every quadratic solution is the quadratic formula:

x = (−b ± √(b² − 4ac)) / (2a)

The formula derives from the technique of completing the square. By substituting the three coefficients a, b, and c, you obtain either two distinct values, one repeated value, or a pair of complex conjugate values for x — the exact outcome depends on the discriminant.

🔍 The Discriminant and Root Classification

The discriminant D = b² − 4ac is the expression under the square root. Its sign alone reveals the nature of the solutions:

D > 0

Two distinct real roots. The parabola crosses the x-axis at two separate points.

D = 0

One repeated real root. The parabola is tangent to the x-axis (touches but does not cross).

D < 0

Two complex conjugate roots. The parabola does not intersect the x-axis at any real point.

Understanding the discriminant before solving saves time — especially on timed exams where recognising an irreducible quadratic (negative discriminant) immediately tells you to switch to complex-number arithmetic.

📍 Vertex, Axis of Symmetry & Parabola Shape

Every parabola has a single turning point called the vertex. Its x-coordinate is the axis of symmetry, given by:

h = −b / (2a)

k = f(h) = a·h² + b·h + c

If a > 0 the parabola opens upward and the vertex is the global minimum. If a < 0 it opens downward and the vertex is the global maximum. The y-intercept is simply f(0) = c — the constant term.

✏️ Vertex Form and Factored Form

Two alternative representations give different geometric insights:

  • Vertex form a(x − h)² + k makes the turning point visible at a glance. It is the preferred form when graphing by hand or identifying transformations of the basic parabola y = x².
  • Factored form a(x − r₁)(x − r₂) shows the x-intercepts directly. It is most useful for root finding, polynomial long division, and understanding the sign of the function between its roots.

🔢 Complex Roots Explained

When D < 0, the square root of a negative number is required. The result is a pair of complex conjugates:

x = (−b / 2a) ± (√|D| / 2a) · i

Complex roots always appear as conjugate pairs when the coefficients are real numbers. They have profound applications in electrical engineering (AC circuit analysis), signal processing (frequency response), control systems, and quantum mechanics.

📊 Reading the Interactive Graph

The parabola graph plots f(x) = ax² + bx + c over a window centred on the vertex. Key visual markers include:

  • Green dots — the real x-intercepts (roots). Absent when roots are complex.
  • Orange dot (V) — the vertex (minimum or maximum).
  • Dashed orange line — the axis of symmetry x = h.

Hover over any point on the curve to see the precise (x, f(x)) coordinates. Try adjusting the coefficients to see how changing a widens or narrows the parabola, changing b shifts it horizontally, and changing cmoves it vertically.

🎓 Step-by-Step Mode

Toggle Show step-by-step to reveal a detailed derivation that walks through every calculation: identifying coefficients, computing the discriminant, applying the quadratic formula, finding the vertex, and deriving the y-intercept. This mode is designed for students who want to verify their work, teachers who need to explain solutions, and professionals who need to audit calculations.

⚡ Practical Applications

Quadratic equations model a remarkable variety of real-world situations:

  • Projectile motion — the height of a ball over time followsh(t) = −½g·t² + v₀t + h₀.
  • Revenue optimisation — profit as a function of price is often quadratic, and the vertex gives the profit-maximising price.
  • Electrical circuits — resonant frequency calculations use quadratic equations involving capacitance and inductance.
  • Geometry — computing areas, distances, and the intersection of geometric objects frequently reduces to a quadratic equation.
  • Computer graphics — ray-sphere intersection tests in rendering engines solve the quadratic that describes a sphere's surface.

✅ Tips for Accurate Results

  • Enter a ≠ 0 — if a = 0 the equation is linear, not quadratic.
  • Use negative values for b and c by typing a minus sign before the number.
  • Adjust decimal precision (0–10) to control how many significant figures appear.
  • For irrational roots (non-perfect-square discriminant), the decimal approximation is shown automatically.
  • Factored form is shown when the roots are rational numbers or simple fractions; otherwise only the vertex form is displayed.

Frequently Asked Questions

Is the Quadratic Equation Solver free?

Yes, Quadratic Equation Solver is totally free :)

Can I use the Quadratic Equation Solver offline?

Yes, you can install the webapp as PWA.

Is it safe to use Quadratic Equation Solver?

Yes, any data related to Quadratic Equation Solver only stored in your browser (if storage required). You can simply clear browser cache to clear all the stored data. We do not store any data on server.

What is a quadratic equation?

A quadratic equation is a polynomial equation of degree two, written in the standard form ax² + bx + c = 0, where a, b, and c are real numbers and a ≠ 0. The term 'quadratic' comes from 'quadratus', Latin for square, referring to the x² term.

How does the Quadratic Equation Solver work?

Enter the coefficients a, b, and c. The solver computes the discriminant D = b² − 4ac, then applies the quadratic formula x = (−b ± √D) / (2a). When D > 0 two distinct real roots are returned; when D = 0 one repeated root is returned; when D < 0 two complex conjugate roots are returned. The tool also computes the vertex, axis of symmetry, factored form, and vertex form.

What does the discriminant tell me?

The discriminant D = b² − 4ac determines the nature of the roots without actually computing them. If D > 0 the parabola crosses the x-axis at two distinct points (two real roots). If D = 0 the parabola is tangent to the x-axis (one repeated real root). If D < 0 the parabola does not cross the x-axis and the roots are complex conjugates.

Can this solver handle complex roots?

Yes. When the discriminant is negative the solver returns complex conjugate roots displayed in a + bi form. For example, for x² + 2x + 5 = 0 the roots are −1 + 2i and −1 − 2i. Complex roots always come in conjugate pairs when the coefficients are real numbers.

What is the vertex form and why is it useful?

The vertex form a(x − h)² + k rewrites the quadratic so the vertex (turning point) coordinates h and k are immediately visible. It is especially useful for graphing, identifying the minimum or maximum value of the function, and solving optimization problems without expanding the expression.

Why must coefficient a be non-zero?

If a = 0 the equation reduces to bx + c = 0, which is a linear equation with at most one solution. A quadratic equation by definition requires the x² term, so a must be a non-zero number. This tool is designed specifically for quadratic equations; use a linear equation solver if a = 0.