Root of Quadratic Equation Calculator

A quadratic equation is a second-degree polynomial equation in a single variable. The general form is ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0. This calculator finds 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, b, and c are constants
a ≠ 0 (if a = 0, the equation becomes linear)
x is the variable

Quadratic equations can represent many real-world situations, such as projectile motion, area problems, and financial calculations. The solutions to a quadratic equation are called roots or solutions.

The Quadratic Formula

The quadratic formula is a standard method for solving quadratic equations. It provides the roots of the equation in terms of its coefficients.

x = [-b ± √(b² - 4ac)] / (2a)

Where:

a, b, and c are the coefficients from the quadratic equation
√(b² - 4ac) is the discriminant
The ± symbol indicates there are two possible solutions

The discriminant (b² - 4ac) determines the nature of the roots:

If discriminant > 0: Two distinct real roots
If discriminant = 0: One real root (repeated)
If discriminant < 0: Two complex conjugate roots

Note: For complex roots, the calculator will display them in the form a ± bi, where i is the imaginary unit (√-1).

How to Use This Calculator

Enter the coefficients a, b, and c in the input fields
Click the "Calculate" button
View the results showing the roots of the equation
Use the "Reset" button to clear the inputs

The calculator will automatically:

Validate the inputs to ensure a is not zero
Calculate the discriminant
Determine the nature of the roots
Display the roots in a clear format

Worked Examples

Example 1: Two Distinct Real Roots

Solve x² - 5x + 6 = 0

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

Example 2: One Real Root (Repeated)

Solve x² - 6x + 9 = 0

Step	Calculation
1	Identify coefficients: a=1, b=-6, c=9
2	Calculate discriminant: (-6)² - 4(1)(9) = 36 - 36 = 0
3	Apply quadratic formula: x = [6 ± √0]/2
4	Calculate root: x = 6/2 = 3 (double root)

Example 3: Complex Roots

Solve x² + 2x + 5 = 0

Step	Calculation
1	Identify coefficients: a=1, b=2, c=5
2	Calculate discriminant: (2)² - 4(1)(5) = 4 - 20 = -16
3	Apply quadratic formula: x = [-2 ± √-16]/2
4	Calculate roots: x = -2/2 ± (4i)/2 = -1 ± 2i

Frequently Asked Questions

What is the difference between a quadratic equation and a linear equation?

A quadratic equation has a term with x², making it a second-degree polynomial. A linear equation has only x terms, making it a first-degree polynomial. Quadratic equations can have up to two solutions, while linear equations have exactly one solution.

How do I know if a quadratic equation has real roots?

A quadratic equation has real roots if the discriminant (b² - 4ac) is greater than or equal to zero. If the discriminant is negative, the roots are complex numbers.

What does it mean if the discriminant is zero?

A discriminant of zero means there is exactly one real root (a repeated root). The parabola touches the x-axis at exactly one point.

Can the quadratic formula be used for any quadratic equation?

Yes, the quadratic formula can solve any quadratic equation of the form ax² + bx + c = 0, provided a ≠ 0. It's a universal method for finding the roots of quadratic equations.

What are complex roots in a quadratic equation?

Complex roots occur when the discriminant is negative, resulting in roots that include the imaginary unit i (√-1). They come in conjugate pairs, like a + bi and a - bi.