Given a set D = {a+b•| a,b ∈ $\mathbb{R}$} and a made-up binary operation, in a quadratic equation.

Question

Given a set D = {a+b•| a,b ∈ $\mathbb{R}$}

And a made-up binary operation on D is defined as follows: (a+b•)(c+d•)= ac+(ad+bc)•

For example, (2+3•)(-3+5•)= (-6+1•) You are not allowed to combine (-6+1•) into -5• because they are not like terms. you are allowed to combine like terms, however, like this: (a+b•)+(c+d•) = a+c+(b+d•)

So the question is: Solve the quadratic equation $x^2$-2x+12•=0

I'm very confused about the 12• and binary operation part. Should the quadratic formula be used here? How would you solve it?

Answer 1

Let's denote the $\bullet$ rather by $q$, even if that doesn't represent any real number as value. So that, $q:=0+1\!\bullet$.

Now we have $q^2=q\cdot q=0+0\bullet=0$, and basically that implies the whole multiplication (just the same way as $i^2=-1$ and linearity generates the multiplication for complex numbers).

We have to solve $x^2-2x+12q=0$. Write up $x$ as $x=a+bq$ then we have $x^2=a^2+2abq$, so what is needed is: $$a^2-2a+(2ab-2b)q=-12q$$ Looking at the 'coordinates' on both sides, we need $a^2-2a=0$ and $2b(a-1)=-12$.