Calculate Quadratic Equation

Quadratic Equation Solver

Enter the coefficients a, b, and c for your quadratic equation in the form ax² + bx + c = 0 to find its roots.

Quadratic equations are fundamental in mathematics, appearing in various fields from physics and engineering to economics and computer science. They are polynomial equations of the second degree, meaning the highest power of the variable is two.

What is a Quadratic Equation?

A standard quadratic equation is expressed in the form:

ax² + bx + c = 0

Where:

• x represents an unknown variable.
• a, b, and c are coefficients, which are real numbers.
• a cannot be equal to zero (if a=0, the equation becomes a linear equation: bx + c = 0).

The solutions to a quadratic equation are called its "roots" or "zeros." These are the values of x that satisfy the equation, making the entire expression equal to zero. Graphically, these roots represent the x-intercepts of the parabola defined by the quadratic function y = ax² + bx + c.

The Quadratic Formula

While some quadratic equations can be solved by factoring, the most universal method for finding the roots of any quadratic equation is the quadratic formula:

x = [-b ± √(b² - 4ac)] / 2a

This formula directly provides the values of x that solve the equation, given the coefficients a, b, and c.

The Discriminant: Unveiling the Nature of Roots

A crucial part of the quadratic formula is the expression under the square root: b² - 4ac. This term is known as the discriminant (often denoted by the Greek letter Delta, Δ). The value of the discriminant tells us about the nature and number of the roots:

1. If b² - 4ac > 0 (Discriminant is positive):

   The equation has two distinct real roots. This means the parabola intersects the x-axis at two different points.

2. If b² - 4ac = 0 (Discriminant is zero):

   The equation has exactly one real root (sometimes called a repeated root or two equal real roots). In this case, the parabola touches the x-axis at exactly one point, its vertex.

3. If b² - 4ac < 0 (Discriminant is negative):

   The equation has two complex conjugate roots. This means the parabola does not intersect the x-axis at all. The roots involve the imaginary unit i, where i = √(-1).

How to Use the Quadratic Equation Solver

Our calculator simplifies the process of finding the roots of any quadratic equation. Follow these steps:

1. Identify Coefficients: Look at your quadratic equation and identify the values for a, b, and c. Remember, if a term is missing, its coefficient is 0 (e.g., in x² + 5 = 0, b=0).
2. Enter Values: Input the numerical values of a, b, and c into the respective fields in the calculator above.
3. Calculate: Click the "Calculate Roots" button.
4. Interpret Results: The calculator will display the roots, indicating whether they are real or complex, and if there are one or two distinct solutions.

Examples:

Example 1: Two Distinct Real Roots

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

• a = 1
• b = -5
• c = 6

Using the calculator with these values will yield:

x₁ = 3.0000

x₂ = 2.0000

Here, the discriminant (-5)² - 4(1)(6) = 25 - 24 = 1, which is positive, indicating two distinct real roots.

Example 2: One Real Root (Repeated)

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

• a = 1
• b = -4
• c = 4

Using the calculator with these values will yield:

x = 2.0000

Here, the discriminant (-4)² - 4(1)(4) = 16 - 16 = 0, which is zero, indicating one real root.

Example 3: Two Complex Conjugate Roots

Consider the equation: x² + x + 1 = 0

• a = 1
• b = 1
• c = 1

Using the calculator with these values will yield:

x₁ = -0.5000 + 0.8660i

x₂ = -0.5000 - 0.8660i

Here, the discriminant (1)² - 4(1)(1) = 1 - 4 = -3, which is negative, indicating two complex conjugate roots.

Conclusion

Quadratic equations are a cornerstone of algebra, and understanding their structure and how to solve them is essential for many scientific and mathematical pursuits. The quadratic formula provides a robust method for finding their roots, and our calculator makes this process quick and error-free, allowing you to focus on interpreting the results and their implications.