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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) : Arithmetics, Mathematics

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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, the discriminant named delta is calculated like this:

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

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 $$ax^2+bx+c$$ has $$\Delta = b^2-4ac = 3^2-4*2*1 = 1$$

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 $$\Delta = 1$$ so $$x_1 = -1/2$$ 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 complex 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.