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."

Related Topics

Quadratic EquationsFactoring & RootsFunction Transformations

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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.
What can it graph?
It can plot explicit, implicit, and parametric functions, add points and geometry, and animate sliders on the same graph.
Can I use voice or a photo?
Yes. You can talk to the tutor, upload a worksheet or handwritten problem, and let the graph update from that input.
Will it explain the steps?
Yes. The AI explains what it is drawing and why, so you see the answer on the graph instead of getting only a final number.