It is not "safe" to assert that a + (bc) = (a + b)(a + b) because, as you demonstrated, doing so necessarily also requires that c = (a^2 + 2ab + b^2 -a)/b. The initial assumption would only be "safe" if there were no way for it to be incorrect.

An equivalent question is "is it correct to assume that x=2x?", and the answer to that equivalent question is no, because there are values of x for which the assumption does not hold

a + (bc) = (a + b)(a + b) — Ulrik

The usual way you deal with bracketed expressions are by axiomatising how products interact with sums and how products interact with products. The sum-product interaction axioms are usually called the distributive laws.

$a.(b+c) = a.b + b.c$
$(b+c).a=b.a+c.a$


There's no need for the second one if the algebra you're dealing with has a commutative product operation, IE where $a.b=b.a$ for all $a$ and $b$. If only the first one holds. the algebra is called 'left distributive', and the first one is usually called 'the left distributive law', when the second one holds, the algebra is called 'right distributive' and the law correspondingly is called 'the right distributive law'.

The product-product interaction axiom is usually called 'associativity of the product operation' or 'the associative law'.

$a.(b.c)=(a.b).c$

When you expand the term $(a+b)(c+d)$, you're covertly using the left distributive law and the right distributive law:

$(a+b)(c+d)=(a+b).c+(a+b).d=a.c+b.c+a.d+b.d$

the first step uses the left distributive law, the second step uses the right distributive law. Structures which have both addition and multiplication operators which interact using a distributive law are typically called 'rings', if the ring also has $a.b=b.a$ for all a and b then it's called a 'commutative ring'. You can look up these topics yourself to see how they are axiomatised (there are more required properties than the ones I've presented here).

a + (bc) = (a + b)(a + b) — Ulrik

For a = b = c = 0, the equation is true.

In mathematics a statement that contains an equals sign can be an identity or an equation.

An identity holds true for all possible values of the pronumerals (variable names). An example is $\sin^2\theta + \cos^2\theta = 1$, which is true for any value of the pronumeral $\theta$.

An equation, when contrasted with an identity, is something that is to be solved in order to find a value for one of the pronumerals. If there is only one pronumeral, the solution will be a number. Otherwise it will be a formula that uses numbers and the other pronumerals. An example is $\sin x = 1$, or another one is wrote you wrote above, followed by the instruction 'solve for c'.

It's hard to tell without a context, but if we don't assume the standard notation and the standard arithmetic, and instead assume some algebra with unknown axioms and notation, then consider an algebra where the following axiom holds:

For any p and q, pq ≡ p

Thanks, I think I get it now. I have shown the equation only holds if c has that particular value. If it has any other value, it doesn't hold. So in general, no, it is not right to assert that

a + (bc) = (a + b)(a + b)



I think the author was referring to identity. Cheers.



Thanks.



Not sure what this means. How to interpret ≡?

Not sure what this means. How to interpret ≡? — Ulrik

It just means equality here. I wrote ≡ to emphasize that the equality holds unconditionally.

For example, in the standard arithmetic p×1 = p for any p other than 0. This equality is axiomatic. But in an alternative algebra it could be axiomatically true that p×q = p for any p and q. In this algebra

a + (bc) = a + b = (a + b)(a + b)