Quadratic Root Formula Calculator
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Reviewed by Calculator Editorial Team
A quadratic equation is a second-degree polynomial equation in a single variable x with three coefficients: a, b, and c. The general form is ax² + bx + c = 0. This calculator helps you find the roots of any quadratic equation using the quadratic formula.
What is a Quadratic Equation?
A quadratic equation is a polynomial equation of degree 2. It has the general form:
ax² + bx + c = 0
Where:
- a is the coefficient of x² (must not be zero)
- b is the coefficient of x
- c is the constant term
The solutions to the equation are called roots or zeros. A quadratic equation can have:
- Two distinct real roots
- One real root (a repeated root)
- No real roots (complex roots)
Quadratic Formula
The quadratic formula is a standard method for finding the roots of any quadratic equation. The formula is:
x = [-b ± √(b² - 4ac)] / (2a)
Where:
- √(b² - 4ac) is called the discriminant
- The discriminant determines the nature of the roots:
| Discriminant | Nature of Roots |
|---|---|
| b² - 4ac > 0 | Two distinct real roots |
| b² - 4ac = 0 | One real root (repeated) |
| b² - 4ac < 0 | No real roots (complex roots) |
Note: The quadratic formula works for all quadratic equations where a ≠ 0. If a = 0, the equation is no longer quadratic.
How to Use This Calculator
- Enter the coefficients a, b, and c of your quadratic equation
- Click the "Calculate Roots" button
- View the results including the roots and discriminant
- Interpret the results based on the discriminant
- Use the chart visualization to understand the relationship between the coefficients and roots
The calculator will display:
- The two roots (if they exist)
- The discriminant value
- An interpretation of the results
- A chart showing the quadratic function
Example Calculation
Let's solve the quadratic equation x² - 5x + 6 = 0 using our calculator.
x² - 5x + 6 = 0
Here, a = 1, b = -5, c = 6.
Using the quadratic formula:
x = [5 ± √(25 - 24)] / 2
x = [5 ± √1] / 2
x = (5 + 1)/2 or (5 - 1)/2
x = 3 or x = 2
The roots are x = 3 and x = 2. The discriminant is 1, indicating two distinct real roots.
Frequently Asked Questions
What is the quadratic formula used for?
The quadratic formula is used to find the roots of any quadratic equation. It's a fundamental tool in algebra and has applications in physics, engineering, and other scientific fields.
What does the discriminant tell me about the roots?
The discriminant (b² - 4ac) determines the nature of the roots. A positive discriminant means two distinct real roots, zero means one real root, and a negative discriminant means two complex roots.
Can the quadratic formula be used for all quadratic equations?
Yes, the quadratic formula can be used for any quadratic equation as long as the coefficient of x² (a) is not zero. If a = 0, the equation is no longer quadratic.
What if the discriminant is negative?
If the discriminant is negative, the equation has no real roots. The roots will be complex numbers. The calculator will display this information in the results.
How accurate is this calculator?
This calculator uses precise mathematical calculations based on the quadratic formula. The results are accurate to the precision limits of floating-point arithmetic in JavaScript.