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Discriminant of a Polynomial

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.

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Discriminant of a Polynomial -

Tag(s) : Functions

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# Discriminant of a Polynomial

## Discriminant Calculator

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.

### 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 polynomialof 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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