The secret to maintaining an equality, or for that matter an inequality, is to treat both parts of the equality the same. Whatever you do to one side of an equation do the same to the other and equality will always be maintained.
Algebra involves unknown numerical values that are represented by letters. Being unknown means you cannot combine them with known values. But you can combine or deduct them from multiple values of themselves. The process of solving algebraic equations involves manipulating unknowns to one side of the equation and knowns to the other.
When learning about quadratic equations we will be told that the solution to an equation of general form ax2 + bx + c = 0 is x = [ -b ± √(b2 -4ac)] /2a
Let’s have a go at proving that
Start by dividing throughout by a so that x2 + bx/a + c/a = 0 and then try to factorise it in the form (x + ?)(x + ?) + some constant = 0.
Clearly to get the term bx/a the ? bit has to be b/2a
The expansion of (x + b/2a)(x + b/2a) = x2 + bx/a + b2/4a2
and if we deduct from both sides of this equation b2/4a2 and add c/a we get
(x + b/2a)(x + b/2a) – b2/4a2 + c/a = x2 + bx/a + c/a = 0
So (x + b/2a)2 = b2/4a2 – c/a and x + b/2a = ± √ ( b2/4a2 – c/a )
which when simplified leads to x = [ -b ± √(b2 -4ac)] /2a