I've been reading a book called Mathematics for the Nonmathematician, and it presents a solution to quadratic equations of the form:
$x^2 + bx + c = 0$
which relies on coming up with a new formula, namely:
$y = x + b/2$
which when rearranged gives:
$x = y - b/2$
which is then substituted into the original formula, giving:
$(y - b/2)^2 + b (y - b/2) + c = 0$
I understand that using $b/2$ makes it easier to simplify the equation.
However, the author also explained that when $b/2$ is added to both of the roots of an equation, the sum of the resulting values is equal to $0$. I am failing to understand the significance/proof of that. I'm not quite sure how the author has come to the conclusion that this will always be the case, and I'm not quite sure whether/why it is important.
I'm a complete maths novice, but a university course I am about to start has a mathematics component, and so I would like to understand as much as I can in an intuitive way before it begins.
The equation $x^2 + bx + c = 0$ is derived from a more general form $ax^2 + bx + c = 0$
The curve $Y = ax^2 + bx + c$ has two characteristics:-
(1) The minimum occurs when $x = \frac {-b}{2a}$ (by assuming a>0);
(2) The axis of symmetry is the vertical line $X = \frac {-b}{2a}$.
Performing the substitution $x’ = x + \frac {-b}{2a}$ is just doing a transformation (translation actually) to the curve by shifting it horizontally left $\frac {-b}{2a}$ units.
After the transformation, the axis of symmetry is then $X = 0$. Such act can make the subsequent works (like roots finding [because they are now symmetrically located about the Y-axis]) a bit easier.
In your case, the ‘a’ is obviously equal to 1.