**A quadratic equation** is an equation of the form: <<a*x^2+b*x+c=0>>

Where *a* is non-zero. Because if *a* is zero, then the equation becomes a linear equation. All coefficients are real numbers.

#### Condition
Let's solve an equation with the following coefficients:

{{{
a = input(1); b = input(7); c = input(-3)
}}}
#### Solution
First of all, we find the discriminant using the following formula:

{{{
D = b^2 - 4*a*c
}}}
The sign of the discriminant determines whether the equation have real roots (if the discriminant is less than zero, then Mathete considers both roots as NaN). If the discriminant is zero, the equation has one root, or in other words, two equal roots.

If the discriminant is greater than zero, then we can find its root

{{{
d = \D
}}}
Now let’s find the roots

{{{
x_1 = (-b-d)/(2*a); x_2 = (-b+d)/(2*a)
}}}
#### Vieta's theorem

Vieta's theorem - formulae expressing the coefficients of a polynomial in terms of its roots. In the case of a quadratic equation, the formulae are:

{{{
<<x_1+x_2=-b/a; x_1*x_2=c/a
}}}
;;Let’s verify for our specific equation

{{{
;;x_1 + x_2 == -b/a; x_1 * x_2 == c/a
}}}
#### Graphical representation

Let's construct a graph of the function <<f(x) = a * x2 + b * x + c>>. The points of intersection of the graph with the x axis will be the roots of the equation.

{{{
;;Delta = (x_2 - x_1+2)/150
;;x_f = (x_1 -1); x_s = x_f + Delta; x = x_f, x_s..(x_2+1); F(t) = a*t^2 + b*t +c
chart({x1: x, y1: F(x), color1: "red", type1: "line", label1: "ax^2+bx+c", x2: ([x[1], x[150]]), y2: ([0, 0]), color2: "green", type2: "line", label2: "", width: "400", height: "250", xaxis: "auto", yaxis: "auto"})
}}}