A quadratic equation is any equation that can be rearranged in standard form as
\[ax^2 + bx + c = 0\,,\]where \(a, b, c \in \mathbb{R}\) and \(a \neq 0\).
The expression
\[\Delta = b^2 - 4ac\]is called the discriminant of the quadratic equation. The sign of \(\Delta\) determines the number of real-valued solutions:
if \(\Delta > 0\), then there are two distinct real-valued solutions (\(x_1 \neq x_2\))
als \(\Delta = 0\), then both real-valued solutions are the same (\(x_1 = x_2\))
als \(\Delta < 0\), then there are no real-valued solutions
The real-valued solutions can be determined as:
\[x_{1} = \frac{-b - \sqrt{\Delta}}{2a}\ \ \ \text{en}\ \ \ x_{2} = \frac{-b + \sqrt{\Delta}}{2a}\]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 \(\Delta\) (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 \(x_2\).
>>> 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)