What Completing the Square Actually Is
Completing the square turns x2+bx into a perfect square by physically filling in the missing corner of a square, then subtracting that same corner back off so the expression's value never changes.
Picture x2 as a square with side x. Add 6x to it, and picture that as two rectangles, each 3 by x, stuck onto two sides of the square. Together the square and the two rectangles almost form a bigger square, but there's a small gap in the corner: a 3-by-3 piece. Drop that 3-by-3 piece in, and you have one clean square of side (x+3), with area (x+3)2. That's the picture behind x2+6x: it's a square missing one corner, and the corner is always (b/2)2 when you start with x2+bx.
But adding that corner piece changes the value of the expression, so you have to subtract it right back off in the same step. That's why x2+6x becomes (x+3)2-9, not just (x+3)2. The -9 is the price of admission: you borrowed a 3-by-3 square to make the picture work, so you pay it back immediately. This is why completing the square isn't just 'halve b, square it, done' — that step only builds the square. The subtraction is the part that keeps the expression honest, and skipping it is the single most common error.
Once an expression sits as (x+h)2+k, a quadratic's structure is exposed. Set y=(x+h)2+k and the vertex of the parabola falls right out: it sits at (-h,k), because (x+h)2 is zero exactly when x=-h, and that's the smallest (or largest) y can get. This connects completing the square directly to vertex form, to the quadratic formula (which is just completing the square done once, in general, on ax2+bx+c), and to solving equations by isolating a squared term.
Key terms
- Completing the square
- Rewriting x2+bx+c as a squared binomial plus a constant by adding and subtracting (b/2)2.
- Vertex form
- A quadratic written as (x+h)2+k, which directly shows the parabola's vertex at (-h,k).
- Perfect square trinomial
- An expression like x2+6x+9 that factors as a single squared binomial, here (x+3)2.
- Vertex
- The highest or lowest point of a parabola, where it turns direction.
Watch it explained
- How it worksHow does splitting 6x into two 3-by-x rectangles and adding a 3-by-3 corner finish the square?
- How it worksWalk me through why the corner piece you just added has to be subtracted again right away.
- Common mistakeWhy isn't completing the square merely a memorized trick of halving b and then squaring it?
- Ties togetherHow does the completed square explain where the vertex of a parabola actually comes from?
Check yourself
End of topic.