Algebra
Completing the Square – Rewriting a Quadratic as (x + d)² + e
Completing the square is a third way to work with a quadratic, alongside factoring and the quadratic formula. It rewrites x² + bx + c in the form (x + d)² + e, where d = b ÷ 2 and e = c − d². Unlike factoring, completing the square always works, even when there's no nice integer factor pair – and as you'll see on the next few pages, it also directly reveals a parabola's vertex.
The name comes from a geometric idea dating back to ancient Babylonian and Greek mathematicians, who solved quadratic problems by literally rearranging areas into a perfect square shape, leaving a small leftover piece to be accounted for – exactly what the “+ e” term represents algebraically today.
Completing the Square as Geometry
The name “completing the square” is not just a figure of speech — it describes an actual shape you can draw. Start with x² + 6x. Picture x² as an x-by-x square, and split 6x into two matching 3-by-x strips, one along the right edge and one along the bottom.
The blue square and two orange strips already cover x² + 6x, but they leave a gap in the corner — a hole of size 3 × 3 = 9. Filling that gap in (the green dashed square) completes a full square of side (x + 3). That's exactly why you add 9 when completing the square for x² + 6x: it's the missing piece needed to make the shape a perfect square, and it's also why the number you add is always (half of b)² — here, half of 6 is 3, and 3² = 9.
Completing the Square Step by Step
For x² + bx + c, the value of d is always b ÷ 2, and e = c − d². The result is (x + d)² + e.
d = 8 ÷ 2 = 4. e = 10 − 4² = 10 − 16 = −6. Result: (x + 4)² − 6.
Real-Life Application
- Optics: completing the square helps locate the focus of a parabolic mirror or dish.
- Engineering: revealing a system's minimum or maximum value directly, without graphing.
- Physics: finding the peak height of a projectile's path without a calculator.
Key Takeaways
- Completing the square rewrites x² + bx + c as (x + d)² + e.
- d = b ÷ 2, and e = c − d².
- This method always works, unlike factoring, which needs a nice integer pair.