Equation Calculator Online

Quadratic Equation Solver

Use this calculator to find the roots of a quadratic equation in the standard form ax² + bx + c = 0.

Quadratic Equation Solver Online: Find Your Roots with Ease

A quadratic equation is a fundamental concept in algebra, widely used across various fields from physics and engineering to economics and finance. It's an equation of the second degree, meaning it contains at least one term where the variable is squared. The standard form of a quadratic equation is:

ax² + bx + c = 0

where x represents an unknown, and a, b, and c are coefficients, with a not equal to zero. If a were zero, the equation would become a linear equation (bx + c = 0).

How to Use This Quadratic Equation Calculator

Our online quadratic equation solver simplifies the process of finding the roots (or solutions) of any quadratic equation. Follow these simple steps:

1. Identify Coefficients: Look at your quadratic equation and identify the values for a, b, and c.
2. Input Values: Enter these numerical values into the respective input fields: "Coefficient a", "Coefficient b", and "Coefficient c".
3. Solve: Click the "Solve Equation" button.
4. View Results: The calculator will instantly display the roots of your equation, along with an explanation of their nature (real, repeated, or complex).

The Quadratic Formula Explained

The calculator uses the well-known quadratic formula to find the roots of the equation. This formula is:

x = [-b ± √(b² - 4ac)] / 2a

The term inside the square root, (b² - 4ac), is called the discriminant (often denoted by Δ). The value of the discriminant determines the nature of the roots:

• If Δ > 0: There are two distinct real roots. This means the parabola (the graph of a quadratic equation) intersects the x-axis at two different points.
• If Δ = 0: There is exactly one real root (also called a repeated root or a double root). The parabola touches the x-axis at exactly one point.
• If Δ < 0: There are two complex conjugate roots. The parabola does not intersect the x-axis at all.

Examples of Quadratic Equations and Their Solutions

Example 1: Two Distinct Real Roots

Consider the equation: x² - 5x + 6 = 0

• Coefficient a = 1
• Coefficient b = -5
• Coefficient c = 6

Using the calculator with these values will yield:

x₁ = 3.0000

x₂ = 2.0000

(Discriminant = (-5)² - 4(1)(6) = 25 - 24 = 1, which is > 0)

Example 2: One Real (Repeated) Root

Consider the equation: x² + 4x + 4 = 0

• Coefficient a = 1
• Coefficient b = 4
• Coefficient c = 4

Using the calculator with these values will yield:

x = -2.0000

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

Example 3: Two Complex Conjugate Roots

Consider the equation: x² + 2x + 5 = 0

• Coefficient a = 1
• Coefficient b = 2
• Coefficient c = 5

Using the calculator with these values will yield:

x₁ = -1.0000 + 2.0000i

x₂ = -1.0000 - 2.0000i

(Discriminant = (2)² - 4(1)(5) = 4 - 20 = -16, which is < 0)

Example 4: Not a Quadratic Equation (a = 0)

Consider the equation: 0x² + 2x + 4 = 0

• Coefficient a = 0
• Coefficient b = 2
• Coefficient c = 4

The calculator will indicate that this is not a quadratic equation, as the coefficient 'a' cannot be zero for a quadratic equation. It simplifies to a linear equation: 2x + 4 = 0, which has a single solution x = -2.

Whether you're a student, engineer, or just curious, this quadratic equation solver is a quick and accurate tool to help you understand and solve these important mathematical problems.