Completing the Square
Some quadratics just won't factor nicely — and when the test asks for the vertex of a parabola or the center of a circle, factoring won't even help. Completing the square is the move that cracks all of them.
The two solutions of x² + 6x + 5 = 0 are x = -5 and x = -1.
y = x² - 4x + 1 in vertex form is (x-2)² - 3; the vertex (lowest point) is (2, -3).
The whole skill is one habit: half the middle, squared, added to both sides.
What are the solutions to x² + -12x + 32 = 0?
Worked examples
What value should be added to x² + 10x to make it a perfect square trinomial?
What are the solutions to x² - 6x - 7 = 0, found by completing the square?
The function f(x) = x² + 8x + 9 has a minimum value at its vertex. What is that minimum value of f(x)?
Common pitfalls
When solving an equation, the (b/2)² you add to the left must also be added to the right, or the equation is no longer balanced. In vertex form (no equals sign), you instead add AND subtract it on the same side.
The (b/2)² rule only works when x² has a coefficient of 1. For 3x² + 12x + ..., factor out the 3 from the x-terms first: 3(x² + 4x) + ..., then complete the square inside the parentheses.
(x - 3)² = 16 gives x - 3 = ±4, not just +4. Forgetting the negative root loses an entire solution — a classic way to pick a wrong 'one-solution' answer.
If b = 5, half is 2.5 and (2.5)² = 6.25 — fractions are fine. Don't round to a 'nicer' number; the perfect square depends on the exact value.
Key takeaways
To complete the square on x² + bx, add (b/2)² — half the middle coefficient, squared.
When solving an equation, add (b/2)² to BOTH sides; then take the square root, keeping ±.
Vertex form (x - h)² + k reveals the vertex (h, k) and the min/max value k of a parabola.
If the x² coefficient isn't 1, factor it out from the x² and x terms first.
Completing the square works on any quadratic, even ones that don't factor nicely.
Try it yourself
5 practice questions on Completing the Square, drawn from the question bank. The tutor is one click away if you get stuck.