Question: The sum of the cubes of two numbers is $2071$, while the sum of the two
numbers themselves is $19$. Find the two numbers.

I've been working hard to solve this problem and I need someone to tell me how to solve it, it perplexes me.

I know the answer is $7$ and $12$ and it might have something to do with $$a^3 + b^3 = (a + b)(a^2 – ab + b^2)$$ but how do they get to that answer?

Any help will be appreciated thank you.

I followed this Yahoo question up to where it said solve it then I searched like $4$ hours looking for a way to solve it, so I'm hoping someone can help me out.

![enter image description here](https://i.stack.imgur.com/SvjD3.png)

Answer: 
Recall the following identity:
$$x^3+y^3 = (x+y)^3 - 3xy(x+y)$$
Hence, we get that
$$2071 = 19^3 - 57xy \implies xy = 84$$
It is now slightly easier to solve
$$x+y=19 \text{ and } xy=84$$
I trust you can finish it off from here. (Hint: Express $y$ as $\dfrac{84}x$ and solve the quadratic $x+\dfrac{84}x = 19$)
Hint: Complete the square. 

$$x^2-xy+y^2 = (x+y)^2 -3xy$$