Convert any quadratic ax² + bx + c into vertex form a(x − h)² + k by completing the square. See every algebraic step, identify the vertex, axis of symmetry, and roots, then visualize the parabola on a coordinate graph.
ax² + bx + c → a(x − h)² + k
To complete the square, we rewrite the quadratic so it contains a perfect square trinomial. The key quantity is the completing term:
(b / 2a)² = b² / (4a²)
We add this to both sides (or inside the factored form) to create the squared binomial (x − h)² where h = −b / (2a).
The constant k = c − b² / (4a) is the value of the function at the vertex.
Vertex form reveals the vertex directly
Vertex form a(x − h)² + k immediately tells you the vertex is at (h, k) — the highest or lowest point of the parabola — without any further calculation.
This is essential for graphing (plot vertex and axis of symmetry first), optimization problems (minimum cost, maximum area), and understanding transformations (shifts and stretches of the base parabola y = x²).
Completing the square is also the foundation of the quadratic formula — the formula is derived by completing the square on the general form ax² + bx + c = 0.
One-on-one Algebra 1 tutoring makes the algebra click — we work through your actual homework problems and build the intuition that carries you through tests and beyond.