Quadratic Formula Calculator – Solve Quadratic Equations and Analyze Parabolas

  • Adam
  • November 4, 2025

Free online quadratic formula calculator to solve quadratic equations instantly. Find roots, discriminant, vertex, axis of symmetry, and analyze parabola properties with step-by-step solutions. Perfect for algebra students and professionals.

Quadratic Formula Calculator

Advanced Tool for Solving Quadratic Equations and Analyzing Parabolas

Quick Navigation

Solve Quadratic Equation

Solve ax² + bx + c = 0

Analyze Quadratic Properties

Get complete analysis of the parabola

What is the Quadratic Formula?

The quadratic formula is a method for solving any quadratic equation of the form ax² + bx + c = 0. It provides the roots (solutions) regardless of whether they are real or complex. The formula is one of the most fundamental tools in algebra and appears throughout mathematics, physics, engineering, and many other fields.

The formula is: x = [-b ± √(b² - 4ac)] / 2a. It uses the coefficients of the equation to calculate exact solutions. The ± symbol indicates there are typically two solutions, one using addition and one using subtraction.

The discriminant (b² - 4ac) is the critical component that determines the nature of the solutions. If it's positive, you get two distinct real roots. If it's zero, you get one repeated real root. If it's negative, you get two complex conjugate roots.

Key Concept: The quadratic formula works for ALL quadratic equations. Even when factoring or other methods fail, the quadratic formula always provides the answer.

Key Features & Capabilities

This comprehensive quadratic calculator provides multiple analysis modes and detailed solutions:

✓ Solve Equations Find roots of any quadratic equation with step-by-step formula application
📊 Complete Analysis Get vertex, axis of symmetry, discriminant, and all quadratic properties
🔢 Real & Complex Roots Handle both real solutions and complex conjugate pairs correctly
📐 Vertex Calculation Find vertex coordinates and convert to vertex form automatically
📋 Discriminant Analysis Understand root nature through discriminant value interpretation
🔄 Axis of Symmetry Calculate and interpret the parabola's line of symmetry
📝 Step-by-Step Solutions See detailed breakdown showing exact calculation process
⬆️ Parabola Direction Determine if parabola opens upward or downward
🎯 Root Verification Confirms each root satisfies the original equation
📋 Multiple Formats Display results in decimal, fraction, and simplified radical forms
📋 Copy to Clipboard One-click copy functionality to transfer results to other applications
📱 Fully Responsive Works seamlessly on desktop, tablet, and mobile devices

How to Use This Calculator

Step-by-Step Guide

  1. Identify the Mode: Choose "Solve" to find roots or "Analyze" for comprehensive parabola analysis including vertex, symmetry, and discriminant.
  2. Get Your Equation in Standard Form: Write your quadratic as ax² + bx + c = 0. If it's not in standard form, rearrange it first.
  3. Identify Coefficients: Extract the values: a (coefficient of x²), b (coefficient of x), and c (constant term). Include signs (negative or positive).
  4. Enter Values: Input a, b, and c into the calculator. Note that a must be non-zero for a true quadratic equation.
  5. Click Solve or Analyze: Press the button to calculate. For "Solve," you get the roots. For "Analyze," you get complete properties.
  6. Review Results: The main result displays the solutions. For "Solve," you see the roots in simplified form.
  7. Study the Breakdown: See detailed steps showing the discriminant calculation, quadratic formula application, and simplification.
  8. Understand Additional Information: The statistics box shows vertex, axis of symmetry, discriminant interpretation, and parabola direction.

Tips for Accurate Use

  • Sign Convention: Include the sign with each coefficient. For x² - 5x + 6 = 0, enter a=1, b=-5, c=6.
  • Standard Form Required: Ensure equation is in ax² + bx + c = 0 form before entering coefficients.
  • Non-Zero a: Verify a ≠ 0. If a = 0, it's not quadratic and the formula doesn't apply.
  • Complex Roots: When discriminant is negative, roots are complex conjugates (involves i = √-1).
  • Verification: Always verify by substituting roots back into the original equation.

Complete Formulas Guide

The Quadratic Formula

Standard Form and Solution
ax² + bx + c = 0

x = [-b ± √(b² - 4ac)] / 2a

Where:
a = coefficient of x² (a ≠ 0)
b = coefficient of x
c = constant term

Example: For x² - 5x + 6 = 0
x = [5 ± √(25 - 24)] / 2 = [5 ± 1] / 2
x = 3 or x = 2

Discriminant

Discriminant and Root Nature
Δ = b² - 4ac (discriminant)

If Δ > 0: Two distinct real roots
If Δ = 0: One repeated real root
If Δ < 0: Two complex conjugate roots

Example: For x² - 5x + 6 = 0
Δ = (-5)² - 4(1)(6) = 25 - 24 = 1 > 0
Therefore: Two distinct real roots

Vertex Coordinates

Finding the Vertex
x-coordinate (axis of symmetry): x = -b / 2a

y-coordinate: y = c - b² / 4a

Or substitute x-coordinate into original equation

Example: For x² - 5x + 6 = 0
x = 5 / 2 = 2.5
y = 6 - 25/4 = 24/4 - 25/4 = -1/4 = -0.25
Vertex: (2.5, -0.25)

Vertex Form

Converting to Vertex Form
y = a(x - h)² + k

Where (h, k) is the vertex

Example: x² - 5x + 6 = 0 becomes
y = 1(x - 2.5)² - 0.25
or: y = (x - 2.5)² - 1/4

Relationship Between Roots and Coefficients

Vieta's Formulas
For roots r and s:
Sum of roots: r + s = -b/a
Product of roots: r × s = c/a

Example: For x² - 5x + 6 = 0
Roots: 2 and 3
Sum: 2 + 3 = 5 = -(-5)/1 ✓
Product: 2 × 3 = 6 = 6/1 ✓

Real-World Applications

Physics - Projectile Motion

The height of a projectile follows a quadratic equation: h(t) = -16t² + v₀t + h₀. Solving using the quadratic formula gives times when the projectile is at ground level or at a specific height.

Business - Break-Even Analysis

Profit functions often form quadratics: P(x) = -x² + 100x - 1000. Finding roots determines break-even points where profit equals zero.

Engineering - Structural Design

Load calculations and stress distributions often involve quadratic relationships. The quadratic formula helps find critical dimensions and maximum loads.

Finance - Investment Returns

Compound interest and growth models sometimes simplify to quadratic equations for optimization and break-even analyses.

Chemistry - Equilibrium Calculations

Chemical equilibrium problems often produce quadratic equations for concentration calculations in certain reaction types.

Statistics - Normal Distribution

Quadratic equations appear in statistical analysis, particularly in variance calculations and distribution analysis.

Worked Examples

Example 1: Simple Quadratic with Two Real Roots

Problem: Solve x² - 5x + 6 = 0

Solution:
Coefficients: a = 1, b = -5, c = 6

Discriminant: Δ = (-5)² - 4(1)(6) = 25 - 24 = 1

x = [5 ± √1] / 2
x = [5 ± 1] / 2

x₁ = (5 + 1) / 2 = 6 / 2 = 3
x₂ = (5 - 1) / 2 = 4 / 2 = 2

Verification:
x = 3: 3² - 5(3) + 6 = 9 - 15 + 6 = 0 ✓
x = 2: 2² - 5(2) + 6 = 4 - 10 + 6 = 0 ✓

Example 2: Quadratic with One Repeated Root

Problem: Solve x² - 4x + 4 = 0

Solution:
Coefficients: a = 1, b = -4, c = 4

Discriminant: Δ = (-4)² - 4(1)(4) = 16 - 16 = 0

x = [4 ± √0] / 2 = 4 / 2 = 2 (repeated root)

This means: (x - 2)² = 0, or x = 2 with multiplicity 2
The parabola touches the x-axis at exactly one point

Example 3: Quadratic with Complex Roots

Problem: Solve x² - 2x + 5 = 0

Solution:
Coefficients: a = 1, b = -2, c = 5

Discriminant: Δ = (-2)² - 4(1)(5) = 4 - 20 = -16

x = [2 ± √(-16)] / 2
x = [2 ± 4i] / 2
x = 1 ± 2i

Two complex conjugate roots: x₁ = 1 + 2i, x₂ = 1 - 2i
The parabola never crosses the x-axis (no real roots)

Example 4: Projectile Motion

Problem: A ball is thrown upward at 48 ft/s from a 64 ft cliff. When does it hit the ground? (h = -16t² + 48t + 64)

Solution:
Set h = 0: -16t² + 48t + 64 = 0
Divide by -16: t² - 3t - 4 = 0

Coefficients: a = 1, b = -3, c = -4
Discriminant: Δ = 9 - 4(1)(-4) = 9 + 16 = 25

t = [3 ± 5] / 2
t = 4 or t = -1

Since time must be positive: t = 4 seconds
The ball hits the ground after 4 seconds

Example 5: Finding Vertex and Axis of Symmetry

Problem: For y = 2x² - 8x + 5, find vertex and axis of symmetry

Solution:
Coefficients: a = 2, b = -8, c = 5

Axis of symmetry: x = -b / 2a = 8 / 4 = x = 2

Vertex y-coordinate: y = 2(2)² - 8(2) + 5
y = 8 - 16 + 5 = -3

Vertex: (2, -3)
Parabola opens upward (a > 0), so vertex is the minimum

Frequently Asked Questions

❓ What's the difference between factoring and the quadratic formula?
Factoring works when the quadratic factors into nice integer or rational factors. The quadratic formula always works for any quadratic. If factoring is difficult or impossible, use the formula. Both give the same answers when factoring is possible.
❓ What does the ± symbol mean?
The ± means you calculate two separate solutions: one with + and one with -. So x = [-b + √Δ] / 2a and x = [-b - √Δ] / 2a. This gives you both roots of the quadratic.
❓ Why is the discriminant important?
The discriminant tells you the nature of solutions before you finish calculating. Positive → 2 real roots. Zero → 1 real root. Negative → 2 complex roots. This helps you understand what to expect.
❓ What are complex roots?
Complex roots involve imaginary numbers (i = √-1). They come in conjugate pairs like a ± bi. While not real numbers, they are valid mathematical solutions to the equation.
❓ Can a parabola have no real roots?
Yes! When discriminant < 0, the parabola never crosses the x-axis. The entire parabola is either above (if a > 0) or below (if a < 0) the x-axis.
❓ What is the vertex?
The vertex is the highest or lowest point of the parabola. If a > 0 (opens upward), vertex is the minimum. If a < 0 (opens downward), vertex is the maximum. It's located at x = -b/2a.
❓ What if my equation isn't in standard form?
Rearrange it to ax² + bx + c = 0 first. Move all terms to one side so the right side equals zero. Then identify a, b, and c and use the calculator.
❓ Can I have a = 0?
No! If a = 0, it's not a quadratic equation—it becomes linear. The quadratic formula specifically requires a ≠ 0. Check your equation and coefficients.
❓ How do I verify my answer?
Substitute each root back into the original equation. If both sides equal zero, the root is correct. Example: For x² - 5x + 6 = 0 with root x = 2: 2² - 5(2) + 6 = 4 - 10 + 6 = 0 ✓
❓ What is axis of symmetry?
The axis of symmetry is the vertical line x = -b/2a that divides the parabola into mirror images. The vertex lies on this line. It's useful for graphing and understanding parabola properties.
❓ Why might roots be complex?
Roots are complex when discriminant < 0. Geometrically, this means the parabola doesn't cross the x-axis, so there are no real solutions. Complex roots still satisfy the equation mathematically.
❓ When would I use this in real life?
Physics (projectile motion), business (break-even analysis), engineering (structural design), finance (investment optimization), chemistry (equilibrium), or any situation modeled by quadratic equations.

Start Solving Quadratic Equations

Whether you're solving algebra equations, analyzing parabolas, modeling physical phenomena, or optimizing business functions, this comprehensive quadratic calculator provides instant solutions with complete analysis. Fast, accurate, and completely free.

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