Converting General to Standard Form

Converting a quadratic function to standard form means completing the square on the function — grouping the x-terms, adding and subtracting the completing constant, and folding the square.

Convert f(x) = x² − 4x + 3. Group the x-terms: (x² − 4x) + 3. Half of −4 is −2; its square is 4. Add and subtract it inside the story: (x² − 4x + 4) + 3 − 4. Fold the perfect square: (x − 2)² − 1. The vertex (2, −1) now reads on sight, and the check runs both ways — expanding (x − 2)² − 1 returns x² − 4x + 3, and the vertex formula agrees: h = 4/2 = 2, k = f(2) = −1. On a function, the balance move changes: there is no other side to add to, so whatever is added is subtracted in the same line, and the function's value never moves.