we are going to create an abstract data type for quadratic equations of the form ax$2+$ bx $+$ c$.$

Here are the functions that operate on quadratics.

make$\_$quadratic$($a$,$ b$,$ c$)--$ This creates and returns a quadratic ax$2+$ bx $+$ c$.$

coefficient$($q$,$ i$)--$ Return the ith coefficient of the quadratic q$.$ That is$,$ if q is ax$2+$ bx $+$ c$,$ then

coefficient$($q$,2)$ is a

coefficient$($q$,1)$ is b

coefficient$($q$,0)$ is c

The two functions above are the only ones that need to know about how quadratics are represented internally. Everything else uses and builds on the two functions above.

These functions perform arithmetic with quadratics.

add$\_$quadratic$($q$1,$ q$2)--$ Takes two quadratics and returns the result of adding them

multiply$($n$,$ q$)--$ Takes a number n and returns the result of multiplying the quadratic by it$,$ i$.$e$.$ n x q $=$ n x $($ax$2+$ bx $+$ c$)$

has$\_$real$\_$roots$($q$)--$ Takes a quadratic q and return true if it has one or two real roots, or false otherwise. Recall from basic algebra that a quadratic has real roots only if the discriminant is zero or positive.

differentiate$\_$quadratic$($q$)--$ Differentiate the quadratic q$,$ i$.$e$.$ perform dq$/$dx$.$ Recall from basic calculus that if q $=$ ax$2+$ bx $+$ c$,$ then dq$/$dx $=2$ax $+$ b

Your task is to write these functions. You can choose any internal representation you wish.