Calculating the roots of a quadratic equation

 The algorithm for calculating the roots of a quadratic equation is based on the quadratic formula. A quadratic equation is typically expressed in the form:

$�{�}^2+��+�=0$

Where:

• $�$, $�$, and $�$ are the coefficients of the equation.
• $�$ represents the variable we're trying to solve for.
• The equation is set to zero.
• Example:
equation:         2x2- 6x+ 4 = 0coefficients:     a = 2, b = -6, c = 4.

To find the roots of the quadratic equation, follow these steps:

1. 1. Input numbers a, b, and c.

2. 2. Calculate the discriminant ($�$): The discriminant is a value derived from the coefficients $�$, $�$, and $�$ and is used to determine the nature of the roots. The discriminant is calculated as:

   $�={�}^2-4��$

3. Determine the nature of the roots:

   • If $�>0$, there are two distinct real roots.
   • If $�=0$, there is one real root (a repeated root or a double root).
   • If $�<0$, there are no real roots (complex roots).

4. 3. Calculate the roots based on the value of the discriminant:

   a. If $�>0$: Two distinct real roots.

   • ${�}_1=\frac{-�+\sqrt{�}}{2�}$
   • ${�}_2=\frac{-�-\sqrt{�}}{2�}$

   b. If $�=0$: One real root.

   • ${�}_1={�}_2=\frac{-�}{2�}$

   c. If $�<0$: No real roots. Complex roots can be expressed as:

   • ${�}_1=\frac{-�}{2�}+\frac{�\sqrt{-�}}{2�}$
   • ${�}_2=\frac{-�}{2�}-\frac{�\sqrt{-�}}{2�}$ Here, $�$ represents the imaginary unit.

These are the fundamental steps for calculating the roots of a quadratic equation. The values of ${�}_1$ and ${�}_2$ represent the roots of the equation. Depending on the discriminant, you'll have real or complex roots, and the values of $�$ can be calculated accordingly.