Completing the Square
Engineer a perfect-square trinomial by adding the exact missing area, then solve or reveal vertex form without changing the equation's truth.
How can an unfinished quadratic expression become one square?
The middle coefficient tells us the missing side length. Half it, square it, and add that value to complete a perfect-square trinomial.
Inside an equation, add the same square-completing value to both sides.
Solve x² + 6x = 7 by completing the square.
The completed binomial will contain x + 3.
Nine is the exact term missing from the perfect square.
Adding 9 to both sides keeps the same solutions.
Both square roots matter, giving x = 1 or x = −7.
If a ≠ 1, divide or factor it out before using the half-and-square rule.
Completing the square is an equivalent transformation, not a one-sided edit.
The plus-or-minus appears when reversing a square, not when building one.