x² Quadratic Formula Calculator

Solve ax² + bx + c = 0 with the quadratic formula. Get discriminant, root type, real or complex roots, vertex, and full step-by-step working.

Quadratic Formula — Both Roots With Discriminant

BrainyCalculators editorial insight — unique to this tool

x = (−b ± √(b²−4ac)) / 2a — discriminant b²−4ac tells nature of roots without solving. Projectile time-of-flight and profit-maximizing price (revenue quadratic in price) use quadratics. Indian JEE problems often test discriminant shortcuts.

When to use this calculator

Use specifically for degree-2 equations. Linear equations → Equation Solver or Algebra.

Need linear equations too?

This page is dedicated to ax² + bx + c = 0. For a tabbed solver covering both linear equations and quadratics, use the Equation Solver →

x = (−b ± √(b² − 4ac)) / (2a)

a (x² coeff.)

b (x coeff.)

c (constant)

What is the Quadratic Formula Calculator?

The quadratic formula calculator is a specialised solver for equations of the form ax² + bx + c = 0, where a is not zero. It uses x = (-b ± √(b² - 4ac)) / 2a to compute both roots and explains whether they are two real roots, one repeated root, or a complex conjugate pair.

This page also reports the discriminant and the vertex of the parabola, which makes it more than a generic root finder. It is best for algebra, graphing, projectile-motion examples, and any problem where the quadratic structure itself matters.

If you need a broader algebra tool that can switch between linear equations and quadratics, use the Equation Solver. This page stays focused on quadratic formula work only.

The Quadratic Formula Explained

Formula: x = (−b ± √(b² − 4ac)) / (2a)

Discriminant: Δ = b² − 4ac

Vertex: x = −b/(2a) y = c − b²/(4a)

How to Solve a Quadratic Equation

1

Write in Standard Form

Rearrange so the equation is ax² + bx + c = 0 with a ≠ 0.
2

Identify Coefficients

Pick out the values of a (x² coeff.), b (x coeff.), and c (constant).
3

Compute the Discriminant

Calculate Δ = b² − 4ac. Its sign determines root type.
4

Apply the Formula

Substitute into x = (−b ± √Δ) / 2a. Use + for x₁, − for x₂.
5

Find the Vertex

x-vertex = −b/(2a); y-vertex = c − b²/(4a). Vertex is the parabola's turning point.

Example — x² − 5x + 6 = 0

a = 1, b = −5, c = 6

Δ = (−5)² − 4(1)(6) = 25 − 24 = 1

Δ > 0 → Two distinct real roots

x₁ = (5 + √1) / 2 = 6/2 = 3

x₂ = (5 − √1) / 2 = 4/2 = 2

Vertex: x = 5/2 = 2.5, y = 6 − 25/4 = −0.25

Check: (x−3)(x−2) = x² − 5x + 6 ✓

How the Quadratic Formula Calculator Works

Formula, assumptions, and calculation steps for this math tool.

Methodology

Math calculators apply the relevant arithmetic, algebraic, geometric, or numeric rule to the values entered and simplify the result where possible.

Calculation Steps

Read the values and operation selected.
Normalize signs, decimals, fractions, or units if needed.
Apply the mathematical rule or formula.
Format the answer and any intermediate values for checking.

Assumptions and Limits

Inputs must be within the supported domain of the operation.
Decimal answers may be rounded for readability.
Symbolic simplification is limited to the calculator scope.

Frequently Asked Questions

What is the quadratic formula?

The quadratic formula x = (−b ± √(b²−4ac)) / (2a) gives the solutions to any quadratic equation ax² + bx + c = 0. It was derived by completing the square and works for all quadratics.

What does the discriminant tell you?

The discriminant Δ = b² − 4ac tells you the nature of the roots before you solve. Δ > 0: two distinct real roots. Δ = 0: one repeated real root. Δ < 0: two complex conjugate roots.

What do complex roots mean?

Complex roots (a ± bi where i = √−1) occur when the parabola does not cross the x-axis. They always come in conjugate pairs. While not real numbers, complex roots are used extensively in engineering and physics.

What is the vertex of a parabola?

The vertex is the highest or lowest point of the parabola y = ax² + bx + c. Its x-coordinate is −b/(2a) and y-coordinate is found by substituting back. If a > 0 the vertex is a minimum; if a < 0 it is a maximum.

Real-World Applications

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Projectile Motion

The height of a projectile at time t is h(t) = −½gt² + v₀t + h₀ — a quadratic equation in t. Setting h(t) = 0 and solving with the quadratic formula gives the time when the projectile hits the ground, from which range and flight time are calculated.

💰

Business Break-Even & Profit Maximisation

Revenue and cost functions are often quadratic — setting profit P(x) = Revenue − Cost = 0 and solving produces the break-even quantity. The vertex of the parabola gives the production level that maximises profit, a fundamental analysis in microeconomics and business planning.

⚡

Electrical Circuit Power

Power dissipated in a resistor is P = I²R — a quadratic relationship between current and power. Finding the current that produces a specific power output requires solving a quadratic equation. Resonance conditions in RLC circuits also produce quadratic characteristic equations.

🏗️

Structural Engineering — Beam Analysis

The bending moment distribution along a simply supported beam under uniform load is a quadratic function of position. Finding where bending moment equals zero (the inflection point) requires solving a quadratic equation — critical for determining reinforcement placement in concrete beams.

📈

Finance — Option Pricing & Risk

Quadratic equations appear in delta-hedging calculations, convexity adjustments for bond price-yield relationships, and various calibration problems in quantitative finance where the relationship between risk factors and price is inherently non-linear.

🌍

Optics — Lens & Mirror Equations

The lens maker's equation and mirror focal length calculations produce quadratic forms when solving for image position or magnification given constraints on focal length and object distance — foundational in camera design, telescope optics, and corrective lens manufacturing.

Common Mistakes

Not rearranging the equation to ax² + bx + c = 0 form first

The quadratic formula requires all terms on one side with zero on the other. An equation like 3x² + 5 = 2x must first be rearranged to 3x² − 2x + 5 = 0 before applying the formula. Applying the formula with incorrect a, b, c values extracted from the un-rearranged equation produces wrong answers.

Sign errors when extracting b

The coefficient b includes its sign. In the equation x² − 5x + 6 = 0, b = −5 (negative). In the formula, −b = −(−5) = +5. A common mistake is treating b as positive when it is negative in the original equation, causing the ±b term to have the wrong sign.

Forgetting to apply the ± to get both roots

The quadratic formula produces two roots: x = (−b + √discriminant) / 2a AND x = (−b − √discriminant) / 2a. Reporting only one root is an incomplete solution — both roots must be found and evaluated to determine which (if either) satisfies the original problem's constraints.

Computing the discriminant incorrectly

The discriminant is b² − 4ac, where the subtraction order matters. Calculating 4ac − b² gives the wrong sign. Also note that 4ac = 4 × a × c — if a = 2, b = 3, c = 1, then 4ac = 4 × 2 × 1 = 8, not 4 × 5 = 20 (a common arithmetic error when evaluating multi-term expressions).

Treating a negative discriminant as "no solution" without checking for complex roots

When b² − 4ac < 0, the square root of a negative number produces complex (imaginary) roots — not "no solution." For real-world physics problems, complex roots typically indicate the physical scenario described cannot occur. But in engineering and signal processing, complex roots (poles and zeros) have significant mathematical meaning and must be reported correctly.

Discriminant Interpretation Quick Reference

Discriminant (b²−4ac)	Root Type	Parabola Behaviour
> 0	Two distinct real roots	Crosses x-axis at two points
= 0	One repeated real root	Touches x-axis at exactly one point (vertex)
< 0	Two complex conjugate roots	Does not cross x-axis
Perfect square	Two rational roots	Factorable over rationals

References

Al-Khwarizmi, M.I.M. Kitāb al-mukhtaṣar fī ḥisāb al-jabr wa-l-muqābala. c. 820 CE.
Stewart, J. Precalculus: Mathematics for Calculus. Cengage, 2016.
Axler, S. Precalculus: A Prelude to Calculus. Wiley, 2017.
Larson, R. and Hostetler, R. Algebra and Trigonometry. Cengage, 2016.
NCTM. Principles and Standards for School Mathematics. National Council of Teachers of Mathematics, 2000.