Can anyone help to explain to me how I would solve this quadratic word problem?

Could you post the original question please? Your question is very difficult to understand... :(

You have to think of this as an algebraic equation.
Let X stand for the smallest of the three consecutive numbers. Then you can write the equation as:

434 = x^2 + (x+1)^2 + (x+2)^2

This will simplify into finding the where this polynomial (on the right) will equal 434 - this can be done in multiple ways...

Hope this helps.

Thank you, however how would go about solving?

If that's it... well, have you simplified the equation?