How do I solve a quadratic equation step by step?

1) Write in standard form ax² + bx + c = 0. 2) Identify a, b, c. 3) Calculate discriminant Δ = b² - 4ac. 4) Apply quadratic formula: x = (-b ± √Δ) / 2a. 5) Simplify the roots.

What is the discriminant and what does it tell us?

The discriminant Δ = b² - 4ac determines the nature of roots. If Δ > 0: two distinct real roots. If Δ = 0: one repeated root. If Δ < 0: two complex conjugate roots.

How do I find the vertex of a parabola?

The vertex (h, k) is found using h = -b/(2a) and k = f(h) = c - b²/(4a). The vertex is the minimum point if a > 0, maximum if a < 0.

How do I factor a quadratic equation?

If roots are r₁ and r₂, the factored form is a(x - r₁)(x - r₂). For x² - 5x + 6 = 0 with roots 2 and 3, factored form is (x - 2)(x - 3).

What are complex roots?

When discriminant Δ < 0, roots are complex: x = (-b ± i√|Δ|) / 2a. Complex roots always come in conjugate pairs like 2+3i and 2-3i.

What is completing the square?

A method to convert ax² + bx + c to vertex form a(x - h)² + k. Add and subtract (b/2a)² inside the equation to create a perfect square.

How do I use the quadratic formula?

Plug coefficients into x = (-b ± √(b² - 4ac)) / 2a. The ± gives two solutions. Example: For 2x² + 5x - 3 = 0, x = (-5 ± √49) / 4 = (-5 ± 7) / 4, so x = 0.5 or x = -3.

What is the axis of symmetry?

The axis of symmetry is a vertical line x = -b/(2a) that passes through the vertex. The parabola is symmetric about this line.

Quadratic Equation Solver

Our free quadratic equation solver helps you find the roots of any quadratic equation using the quadratic formula. Get the discriminant, vertex form, axis of symmetry, factored form, and complete step-by-step solutions. Works with real and complex roots.

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Equation

1x² - 5x + 6 = 0

Coefficients ax² + bx + c = 0

a (x²)

b (x)

Discriminant (Δ)

2 Real Roots

Δ = (-5)² - 4(1)(6) = 25 - 24 = 1

Quadratic Formula

x = (-b ± √Δ) / 2a

Solutions

x₁

x₂

Properties

Vertex (h, k) (2.5, -0.25)

Axis of Symmetry x = 2.5

Sum of Roots 5

Product of Roots 6

Forms

Factored: (x - 3)(x - 2)

Vertex: (x - 2.5)² - 0.25

Solution Steps

1. Δ = b² - 4ac = 25 - 24 = 1

2. x = (-(-5) ± √1) / 2(1)

3. x = (5 ± 1) / 2

4. x₁ = 3, x₂ = 2

auto_awesome AI Analysis

Quadratic: x² - 5x + 6 = 0

Solutions: x = 3 and x = 2
Factored: (x - 3)(x - 2) = 0
Vertex: (2.5, -0.25) - minimum point
Discriminant: 1 (positive = 2 real roots)

Check: (3)² - 5(3) + 6 = 9 - 15 + 6 = 0 ✓

lightbulb Tips

•x = (-b ± √(b²-4ac)) / 2a
•Δ > 0: two real roots
•Δ = 0: one repeated root
•Δ < 0: complex roots

functions Reference

Discriminant (Δ = b²-4ac)

Δ > 0 Two real roots

Δ = 0 One repeated root

Δ < 0 Complex roots

Formulas

Vertex: (-b/2a, f(-b/2a))

Sum of roots: -b/a

Product of roots: c/a

How to Use This Calculator

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Enter Coefficients

Input values for a, b, and c in ax² + bx + c = 0.

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View Roots

See the solutions x₁ and x₂ (real or complex).

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Analyze Equation

Get discriminant, vertex, axis of symmetry, and forms.

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Follow Steps

Learn the complete solution process step by step.

The Formula

The quadratic formula gives the solutions to any quadratic equation. The discriminant (Δ) determines whether roots are real (Δ ≥ 0) or complex (Δ < 0).

x = (-b ± √(b² - 4ac)) / 2a

lightbulb Variables Explained

a, b, c Coefficients in ax² + bx + c = 0
Δ = b² - 4ac Discriminant (determines nature of roots)
x₁, x₂ The two roots (solutions)

tips_and_updates Pro Tips

If Δ > 0: Two distinct real roots

If Δ = 0: One repeated real root (perfect square)

If Δ < 0: Two complex conjugate roots

Vertex form: a(x - h)² + k where h = -b/2a

Sum of roots = -b/a, Product of roots = c/a

The parabola opens upward if a > 0, downward if a < 0

Quadratic Equation Solver - Find Roots Instantly

Solve any quadratic equation ax² + bx + c = 0 using the quadratic formula. Get real or complex roots, discriminant, vertex, axis of symmetry, and detailed step-by-step solutions.

Quadratic Formula Calculator

Apply the quadratic formula x = (-b ± √(b² - 4ac)) / 2a to find exact solutions. Works for any coefficients with complete step-by-step explanation.

Discriminant Calculator

Calculate the discriminant Δ = b² - 4ac to determine the nature of roots before solving. Positive means two real roots, zero means one root, negative means complex roots.

Vertex Form Calculator

Convert standard form to vertex form a(x - h)² + k. Find the vertex coordinates and understand the parabola's position and direction.

Factoring Quadratic Calculator

Get the factored form a(x - r₁)(x - r₂) when roots are found. Useful for understanding the equation's structure and solving inequalities.

Complex Roots Solver

When the discriminant is negative, find complex conjugate roots in the form a ± bi. Our solver handles both real and imaginary solutions.

Step-by-Step Solutions

Learn how to solve quadratic equations with detailed explanations. Perfect for students studying algebra and preparing for exams.

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