## Discriminant Calculator

## Answers to Questions

### How to calculate a discriminant?

For a **quadratic polynomial** $ ax^2+bx+c $, the **discriminant** named delta $ \Delta $ is calculated with the formula:

$$ \Delta = b^2-4ac $$

The fact of knowing the value of the **discriminant** then solves the equation more easily through formulas (using this **discriminant**).

Example: The equation $ 2x^2+3x+1 = 0 $ of type $ ax^2+bx+c $ (with $ a = 2 $, $ b = 3 $ et $ c = 1 $) has for **discriminant** $ \Delta = b^2-4ac = 3^2-4*2*1 = 1 $

For a **cubic polynomial**of the form \(ax^3+bx^2+cx+d\) the **discriminant** formula is

$$ \Delta = b^2c^2-4ac^3-4b^3d-27a^2d^2+18abcd $$

For a **polynomial of degree 1 or 0** the determinant is not generally calculated, its value has no interest.

### How to find the roots of a polynomial with the determinant?

For a quadratic polynomial of type $ ax^2+bx+c = 0 $

If the **discriminant** is positive (strictly), the equation has two solutions:

$$ x_1 = \frac {-b + \sqrt \Delta}{2a} \\ x_2 = \frac {-b - \sqrt \Delta}{2a} $$

Example: The equation $ 2x^2+3x+1 = 0 $ has for **discriminant** $ \Delta = 1 $, so solutions are $ x_1 = -1/2 $ and $ x_2 = -1 $

If the **discriminant** is zero, the equation has a double root:

$$ x_1=x_2 = -\frac b{2a} $$

If the **discriminant** is negative (strictly), the equation has 2 complex conjugate solutions:

$$ \delta^2 = \Delta $$

$$ x_1 = \frac {-b + \delta}{2a} \\ x_2 = \frac {-b - \delta}{2a} $$

For equations of higher degrees, the calculations are much more complicated, but knowledge of the determinants is important.

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