Functions
AI Assistant

Completing the Square

Transform standard form into vertex form to reveal the vertex

Completing the square is a technique that rewrites ax² + bx + c into vertex form a(x − h)² + k. In vertex form, you can read the vertex (h, k) directly — no formula needed.

The key idea: take the coefficient of x, halve it, and square it. For x² + 6x, half of 6 is 3, and 3² = 9. Add and subtract 9 to get (x + 3)² − 4 when the constant is 5.

This lesson shows the process visually on the graph. Ask the AI anything — try "Why does this work?" or "Complete the square for x² − 4x + 1."

Graph

FAQ

What does completing the square mean?
Completing the square means rewriting a quadratic expression so it contains a perfect square trinomial. You transform x² + bx + c into (x + b/2)² + (c − b²/4). This reveals the vertex of the parabola.
Why not just use the vertex formula?
The vertex formula x = −b/(2a) actually comes FROM completing the square. Understanding the technique gives you deeper insight into why the formula works, and it is essential for deriving the quadratic formula itself.
When do I need completing the square?
You need it to convert standard form to vertex form, to derive the quadratic formula, to solve certain integrals in calculus, and to rewrite circle/ellipse equations into standard form. It is one of the most useful algebraic techniques.
What if a is not 1?
Factor out a from the first two terms first: ax² + bx + c = a(x² + (b/a)x) + c. Then complete the square inside the parentheses and adjust the constant.

Related Topics

Quadratic EquationsFactoring & RootsFunction Transformations