A quadratic equation is any equation that can be rearranged in standard form as

where $a,b,c\in \mathbb{R}$ and $a\ne 0$.

The expression

is called the discriminant of the quadratic equation. The sign of $\mathrm{\Delta }$ determines the number of real-valued solutions:

• if $\mathrm{\Delta }>0$, then there are two distinct real-valued solutions ($x_1\ne x_2$)

• als $\mathrm{\Delta }=0$, then both real-valued solutions are the same ($x_1=x_2$)

• als $\mathrm{\Delta }<0$, then there are no real-valued solutions

The real-valued solutions can be determined as:

Assignment

• Write a function discriminant that takes the three parameters $a$, $b$ and $c$ (int or float) of a quadratic equation. The function must return the discriminant $\mathrm{\Delta }$ (float) of the quadratic equation.

• Write a function solutions that takes the three parameters $a$, $b$ and $c$ (int or float) of a quadratic equation. The function must return three values: i) the number of different real-valued solutions (int) of the quadratic equation, ii) the solution $x_1$ (float) of the quadratic equation and iii) the solution $x_2$ (float) of the quadratic equation. If the quadratic equation has no real-valued solution, the value $0$ must be returned for both $x_1$ and $x_2$.

Example

                
            >>> discriminant(1, 0, -1)
4.0
>>> discriminant(1, 4, -5)
36.0

>>> solutions(1, 0, -1)
(2, -1.0, 1.0)
>>> solutions(1, 4, -5)
(2, -5.0, 1.0)