Completing the Square

Completing the square turns any quadratic into the square-root method.

For x² + 6x − 7 = 0:
1. Move the constant: x² + 6x = 7.
2. Take half of 6 (that is 3), square it (that is 9), add to both sides: x² + 6x + 9 = 16.
3. The left side is now (x + 3)²: (x + 3)² = 16.
4. Take roots: x + 3 = ±4, so x = 1 or x = −7.

The number you add is always (b/2)². Half, then square.

If the leading coefficient is not 1, divide the whole equation by it first.

The technique constructs the perfect square trinomial x² + bx + (b/2)² = (x + b/2)². Adding (b/2)² to both sides preserves equality, and the left side becomes a square by design.

When a ≠ 1, divide through by a first (or factor a out of the two variable terms), since the pattern is stated for a monic quadratic.

Its importance is structural rather than computational. Applying it to the general equation ax² + bx + c = 0 produces the quadratic formula:

x = (−b ± √(b² − 4ac))/(2a)

so the formula is not an independent fact but this procedure carried out once symbolically.

It is also the standard route from standard form to vertex form a(x − h)² + k, which is why it is required for graphing, and the same completion appears in Algebra 2 when converting conic sections into their centre-radius forms.