📐 Quadratic Equation Solver

Quadratic Equation Solver

Solve any quadratic equation: ax² + bx + c = 0. Get real and complex roots, discriminant, vertex, and parabola visualization. Perfect for students, teachers, and math enthusiasts.

a≠0
Leading Coefficient
Δ
Discriminant
2
Max Roots
📈
Parabola Graph

📐 Quadratic Equation Solver: ax² + bx + c = 0

x² - 5x + 6 = 0

Solution & Analysis

Discriminant (Δ) 1
Nature of Roots Two distinct real roots
Vertex (2.5, -0.25)
Roots (Solutions) x = 2, x = 3

Step-by-Step Solution

Step 1: Identify coefficients: a = 1, b = -5, c = 6
Step 2: Calculate discriminant: Δ = b² - 4ac = (-5)² - 4(1)(6) = 25 - 24 = 1
Step 3: Apply quadratic formula: x = [-b ± √Δ] / (2a)
Step 4: x₁ = [5 + 1] / 2 = 3, x₂ = [5 - 1] / 2 = 2

📈 Parabola Graph (y = ax² + bx + c)

The parabola opens upward. Vertex is the minimum/maximum point.

Understanding Quadratic Equations & The Quadratic Formula

A quadratic equation is any equation that can be written in the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. The name "quadratic" comes from "quadratus" (Latin for square), referring to the x² term. Quadratic equations appear in physics (projectile motion), engineering, economics (profit maximization), biology (population models), and everyday problem-solving.

The quadratic formula is the universal solution: x = [-b ± √(b² - 4ac)] / (2a). This formula works for every quadratic equation, whether the roots are real or complex. The expression under the square root, Δ = b² - 4ac, is called the discriminant — it tells us the nature of the roots:

  • Δ > 0: Two distinct real roots (parabola crosses x-axis at two points)
  • Δ = 0: One repeated real root (parabola touches x-axis at vertex)
  • Δ < 0: Two complex conjugate roots (parabola never touches x-axis)

The calculator also finds the vertex of the parabola — the highest or lowest point. Vertex x-coordinate = -b/(2a). This is where the parabola reaches its minimum (if a > 0) or maximum (if a < 0).

📘 How to Use This Quadratic Solver

  1. Enter coefficients a, b, c for your equation ax² + bx + c = 0
  2. Click "Solve Equation" to see results
  3. View the discriminant, root nature, and actual roots
  4. Review the step-by-step solution using the quadratic formula
  5. See the parabola graph visualizing the equation
  6. Use the vertex to understand the parabola's shape

💡 Pro Tip:

If a = 0, the equation becomes linear (bx + c = 0). For linear equations, use our linear equation solver instead.

📊 Real-World Example: Projectile Motion

Problem: A ball is thrown upward with initial velocity 20 m/s from height 5m. Height equation: h = -4.9t² + 20t + 5. When does the ball hit the ground?

Equation: -4.9t² + 20t + 5 = 0

Solution using quadratic formula:

a = -4.9, b = 20, c = 5

Δ = 20² - 4(-4.9)(5) = 400 + 98 = 498

t = [-20 ± √498] / (2 × -4.9) = 4.28 seconds (positive root)

The ball hits the ground after approximately 4.28 seconds. The quadratic solver instantly gives both roots!

📊 Discriminant (Δ) Guide: What It Tells You

Δ > 0

Two Real Roots

Parabola crosses x-axis at 2 points

Example: x² - 5x + 6 = 0 → x = 2, 3

Δ = 0

One Real Root (Double Root)

Parabola touches x-axis at vertex

Example: x² - 4x + 4 = 0 → x = 2 (double)

Δ < 0

Two Complex Roots

Parabola never touches x-axis

Example: x² + x + 1 = 0 → complex roots

📐 Vertex & Parabola Properties

Vertex Formula

xᵥ = -b/(2a)

yᵥ = f(xᵥ) = a(xᵥ)² + b(xᵥ) + c

The vertex is the minimum point (a > 0) or maximum point (a < 0)

Axis of Symmetry

x = -b/(2a)

The vertical line through the vertex; parabola is symmetric about this line

y-intercept

Set x = 0 → y = c

The point where parabola crosses the y-axis

Opening Direction

a > 0 → Opens Upward (U-shaped, minimum)

a < 0 → Opens Downward (∩-shaped, maximum)

Alternative Method: Factoring Quadratics

For simple quadratics where a = 1, factoring is faster. Look for two numbers that multiply to c and add to b. Example: x² - 5x + 6 = 0 → factors: (x - 2)(x - 3) = 0 → x = 2, 3. If a ≠ 1, use the quadratic formula — it always works!

x² + 7x + 12 = 0 → (x+3)(x+4) → x = -3, -4
x² - 9 = 0 → (x-3)(x+3) → x = 3, -3
x² - 2x - 15 = 0 → (x-5)(x+3) → x = 5, -3
x² - 6x + 9 = 0 → (x-3)² → x = 3 (double)

⚠️ 5 Common Quadratic Equation Mistakes

  • Forgetting the ± sign — The quadratic formula gives TWO roots unless Δ = 0
  • Misidentifying a, b, c — Always write in standard form ax² + bx + c = 0 first
  • Incorrect sign when b is negative — -b becomes positive: careful with signs!
  • Dividing incorrectly in the formula — Remember denominator is 2a, not just 2
  • Ignoring complex roots — When Δ < 0, roots are complex (a ± bi). Our calculator shows them!

💡 Verification Tip: Always plug your roots back into the original equation to verify they satisfy ax² + bx + c = 0.

❓ Frequently Asked Questions (Quadratic Solver)

1. What is the quadratic formula?

x = [-b ± √(b² - 4ac)] / (2a). It solves any quadratic equation ax² + bx + c = 0, giving real or complex roots.

2. What does the discriminant tell me?

Δ = b² - 4ac. If Δ > 0: 2 real roots; Δ = 0: 1 real root (double); Δ < 0: 2 complex roots.

3. How do I find the vertex of a parabola?

xᵥ = -b/(2a); yᵥ = a(xᵥ)² + b(xᵥ) + c. Vertex is the minimum (a > 0) or maximum (a < 0) point.

4. Can a quadratic have only one solution?

Yes, when discriminant = 0. The equation has one repeated root (double root). Example: x² - 4x + 4 = 0 → x = 2 (double).

5. What happens if a = 0?

The equation becomes linear (bx + c = 0). Our calculator requires a ≠ 0 for quadratic. For linear equations, x = -c/b.

6. How do I get complex roots?

When Δ < 0, roots are complex conjugates: x=[-b ± i√|Δ|] / (2a). Example: x² + x + 1=0 → x=-0.5 ± 0.866i.

7. Why do I get an error when a = 0?

The quadratic formula divides by 2a. If a = 0, the equation is linear, not quadratic. Reset a to a non-zero value.

8. Is my data stored or shared?

Never. All calculations happen locally in your browser. ToolHub does not track, store, or transmit any data — complete privacy guaranteed.

🔗 More Math & Calculator Tools

Scientific Calc Percentage Calc Fraction Calc Ratio Calc Std Deviation

Explore 75+ free tools — all private, no sign-up, always free.

⚠️ Disclaimer: This quadratic equation solver provides accurate mathematical solutions for educational purposes. Verify critical calculations independently.

🔒 100% private — no data storage. ToolHub does not collect or share your calculations.