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Discriminant of a Polynomial

Tool to compute the discriminant of a polynomial to deduce its roots (values or the expression is zero, equal to 0).

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Discriminant of a Polynomial -

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# Discriminant of a Polynomial

## Discriminant Calculator

### Root Calculator

⮞ Go to: Polynomial Root

### What is a discriminant? (Definition)

A discriminant of a polynomial is an expression giving information about the number and 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 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 discriminant is not generally calculated, its value has no interest, it is nevertheless sometimes defined at the value $1$.

### What is the link between the discriminant and the roots of a polynomial?

The discriminant acts as an indicator of whether the polynomial has real, distinct or imaginary roots and their number.

For a polynomial of degree 2 if the discriminant is equal to $0$, then the polynomial has a unique (double) root. If the discriminant is positive, then it has 2 real roots, and if it is negative, it has 2 complex and conjugate roots.

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

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

If the discriminant is positive (strictly), the equation has two solutions x1 and x2:

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

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 (degree 3 or 4 or more), knowledge of the discriminant allows us to know the number of roots, however there is no formula allowing them to be found from the discriminant.

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