Tool to compute the discriminant of a polynomial. A discriminant of a polynomial is an expression giving information about the nature of the roots of the polynomial.
Discriminant of a Polynomial - dCode
Tag(s) : Functions
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Discriminant of a Polynomial
Discriminant Calculator
Answers to Questions (FAQ)
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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