How this instrument works
Completing the square rewrites a quadratic equation, ax² + bx + c = 0, into the form (x + p)² = q, isolating a perfect square on one side. Dividing through by a, moving the constant to the other side, and adding the square of half the linear coefficient to both sides turns the left side into that perfect square automatically: p = b ⁄ 2a, and q = p² − c⁄a is what remains on the right once that addition is accounted for.
Once in this form, solving for x is a single step: take the square root of both sides and subtract p, giving x = −p ± √q — the same two roots the quadratic formula produces directly, but reached through a method that shows the intermediate reshaping rather than jumping straight to a memorized formula.
This technique is worth knowing beyond just solving equations: it's the same process behind deriving the quadratic formula itself in the first place, and it's the identical move used to convert a quadratic from standard form into vertex form, y = a(x−h)² + k — the vertex (h, k) is simply (−p, c − ap²) once p and q are known here.
- Enter the quadratic's leading coefficient into the a field.
- Enter the linear coefficient into the b field.
- Enter the constant term into the c field.
- Read p and q: the equation completes to (x + p)² = q, ready to solve by taking a square root.
Worked example — x² + 6x + 5 = 0
x² + 6x + 5 = 0 completes the square to (x + 3)² = 4 — half of 6 is 3, so p = 3, and q = 3² − 5 = 4. Taking a square root of both sides gives x + 3 = ±2, so x = −1 or x = −5, matching what the quadratic formula would give directly for these same coefficients.
2x² + 8x + 6 = 0 first divides through by a = 2, giving x² + 4x + 3 = 0, then completes to (x + 2)² = 1 — half of 4 is 2, so p = 2, and q = 4 − 3 = 1. Solving x + 2 = ±1 gives x = −1 or x = −3.
Questions
What does 'completing the square' mean?
It means rewriting a quadratic equation into the form (x+p)² = q by adding a carefully chosen constant to both sides — one that turns the left side into a perfect square trinomial, ready to solve by taking a square root.
How do you find p and q?
p is half of b divided by a (p = b ⁄ 2a), and q is p squared minus c divided by a (q = p² − c⁄a). These come directly from dividing the original equation by a, moving the constant aside, and adding the square needed to complete the perfect-square trinomial.
How is this different from the quadratic formula?
Both reach the identical two roots, but the quadratic formula jumps straight to x = (−b ± √(b²−4ac)) ⁄ 2a as a single memorized formula, while completing the square shows the intermediate reshaping step by step — in fact, deriving the quadratic formula from scratch uses exactly this completing-the-square process.
What does q being negative mean?
It means the original equation has no real solutions — taking the square root of a negative q has no real answer, matching the case where the quadratic formula's own discriminant, b²−4ac, comes out negative too.
Is this related to vertex form?
Yes — converting standard form to vertex form, y = a(x−h)² + k, uses the identical completing-the-square process. The vertex coordinates work out to (−p, c − ap²) once p and q from this page are already known.