Roots in Quadratic Equations

Yaser Rahmati | یاسر رحمتی

A second-order equation, also known as a quadratic equation, is typically written in the form:

$$ax^2 + bx + c = 0$$

To find the roots of such an equation, you can use the quadratic formula:

$$x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}$$

Here's a step-by-step guide on how to use the quadratic formula:

Identify the coefficients:

In the equation $ax^2 + bx + c = 0$, $a$, $b$, and $c$ are the coefficients. $a$, $b$, and $c$ can be any real numbers, including zero.

Plug the coefficients into the quadratic formula:

Once you have identified the coefficients $a$, $b$, and $c$, plug them into the quadratic formula:

$x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}$

Calculate the discriminant:

The discriminant, $b^2 - 4ac$, is part of the quadratic formula. It determines the nature of the roots. There are three cases:

• If $b^2 - 4ac>0$, the quadratic equation has two distinct real roots.

• If $b^2 - 4ac=0$, the quadratic equation has one real root (a repeated root).

• If $b^2 - 4ac<0$, the quadratic equation has two complex roots.

Example

Let's say we have the equation $2x^2−5x+2=0$.

• Identify the coefficients: $a=2$, $b=−5$, and $c=2$.

• Plug the coefficients into the quadratic formula:

$$x = \frac{{- (-5) \pm \sqrt{{(-5)^2 - 4(2)(2)}}}}{{2(2)}} \\$$

• So, the roots are:

$$x_1 = \frac{5+3}{4} = \frac{8}{4} = 2$$

$$x_2 = \frac{5-3}{4} = \frac{2}{4} = 0.5$$

Python Code

This code defines a function find_roots that takes the coefficients a, b, and c of the quadratic equation $ax^2+bx+c=0$ as input and returns the roots.

Depending on the value of the discriminant, it calculates real roots, repeated roots, or complex roots accordingly.

You can replace the values of a, b, and c with your own coefficients to find the roots of any specific quadratic equation.

import math

def find_roots(a, b, c):
    discriminant = b**2 - 4*a*c
    
    if discriminant > 0:
        root1 = (-b + math.sqrt(discriminant)) / (2*a)
        root2 = (-b - math.sqrt(discriminant)) / (2*a)
        return root1, root2
    elif discriminant == 0:
        root = -b / (2*a)
        return root,
    else:
        real_part = -b / (2*a)
        imaginary_part = math.sqrt(abs(discriminant)) / (2*a)
        return (real_part + imaginary_part * 1j), (real_part - imaginary_part * 1j)

# Example usage:
a = 2
b = -5
c = 2
roots = find_roots(a, b, c)
print("Roots:", roots)

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