Roots and Coefficient of Quadratic Equation Calculator

Quadratic equations are fundamental in algebra and appear in many real-world applications. This calculator helps you find the roots (solutions) of a quadratic equation and determine its coefficients when given the roots.

What is a Quadratic Equation?

A quadratic equation is a second-degree polynomial equation in a single variable x with the general form:

ax² + bx + c = 0

Where:

a, b, and c are constants
a ≠ 0 (if a = 0, the equation is linear, not quadratic)
x is the variable

Quadratic equations can have two real roots, one real root (a repeated root), or no real roots (complex roots).

Standard Form of a Quadratic Equation

The standard form of a quadratic equation is:

ax² + bx + c = 0

This form is useful for:

Identifying the coefficients a, b, and c
Applying the quadratic formula to find roots
Graphing the quadratic function

When given the roots of a quadratic equation, you can find the standard form using the factored form.

The Quadratic Formula

The quadratic formula allows you to find the roots of any quadratic equation:

x = [-b ± √(b² - 4ac)] / (2a)

Where:

a, b, and c are the coefficients from the standard form
√(b² - 4ac) is the discriminant
The ± indicates there are two roots

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

If b² - 4ac > 0: Two distinct real roots
If b² - 4ac = 0: One real root (repeated)
If b² - 4ac < 0: Two complex conjugate roots

Using the Calculator

This calculator can work in two modes:

Given coefficients a, b, and c, find the roots
Given two roots, find the coefficients a, b, and c

Simply enter the known values and click "Calculate" to see the results.

Worked Examples

Example 1: Finding Roots

Given the quadratic equation: 2x² + 5x - 3 = 0

Using the quadratic formula:

x = [-5 ± √(5² - 4*2*(-3))] / (2*2)

x = [-5 ± √(25 + 24)] / 4

x = [-5 ± √49] / 4

x = [-5 ± 7] / 4

So the roots are:

x = (-5 + 7)/4 = 2/4 = 0.5
x = (-5 - 7)/4 = -12/4 = -3

Example 2: Finding Coefficients

Given the roots x = 2 and x = -1, find the coefficients a, b, and c.

The factored form is: a(x - 2)(x + 1) = 0

Expanding: a(x² - x - 2) = 0

Which gives: ax² - ax - 2a = 0

Comparing with standard form ax² + bx + c = 0:

a = a (same coefficient)
b = -a
c = -2a

If we choose a = 1, then b = -1 and c = -2.

FAQ

What is the difference between a quadratic equation and a linear equation?: A quadratic equation has an x² term, while a linear equation has only x terms. 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?: Check the discriminant (b² - 4ac). If it's positive, there are two real roots. If zero, one real root. If negative, no real roots (complex roots).
Can I use this calculator for higher-degree polynomials?: No, this calculator is specifically for quadratic equations (degree 2). For higher-degree polynomials, you would need a different tool.
What if I enter a = 0 in the calculator?: The calculator will show an error because a quadratic equation requires a ≠ 0. The equation would become linear instead.
How accurate are the calculations?: The calculator uses standard floating-point arithmetic, which is accurate to about 15 decimal places. For most practical purposes, this is sufficient.