Completing the Square

Master completing the square method for solving quadratics. Math 20-1 Alberta.

Lesson 1.8 of Quadratic Functions & Equations in Math 20-1 — Alberta curriculum lessons.

What you'll learn in this lesson

• Why We Need Completing the Square. Compare general form and vertex form side by side, see a calculator example reading all parabola properties, and learn the easy direction of conversion: expanding vertex form into general form.
• Completing the Square (when a = 1). Learn the perfect-square-trinomial trick and apply it to convert quadratics from general form to vertex form when the leading coefficient is 1.
• Completing the Square (when a ≠ 1). Handle the general case by factoring the leading coefficient out of the first two terms, then complete the square — including examples with negatives and fractions.
• Solving Quadratic Equations by Completing the Square. Apply completing the square to solve equations that cannot be factored, build quadratic equations from irrational roots, and choose the right method for any situation.

Inside the lesson — a free preview

The Two Forms of a Quadratic — A Comparison

You have seen quadratics in two forms. Let's compare what each tells you at a glance.

Why Vertex Form Wins. Vertex form is much more useful for understanding the parabola — it tells you the vertex, axis of symmetry, and maximum or minimum value directly from the equation.

So we need a way to convert from general form to vertex form. That method is called completing the square.

The Easy Direction: Vertex Form to General Form

The Easy Direction. Converting from vertex form to general form is straightforward: just expand and simplify.

Example 2 — Convert y = -3(x + 1)^2 - 5 to General Form. Step 1 — Expand the squared bracket.

(x + 1)^2 = (x + 1)(x + 1) = x^2 + 2x + 1

Step 2 — Substitute back in.

y = -3(x^2 + 2x + 1) - 5

Step 3 — Distribute -3.

y = -3x^2 - 6x - 3 - 5

Step 4 — Combine like terms.

y = -3x^2 - 6x - 8

Properties you can now read directly:

y-intercept (the constant): -8

Answer: y = -3x^2 - 6x - 8

Worked example: Reading Properties from a Calculator

Setup. For y = -(1)/(3)x^2 + 6x + 2, find all the properties using a graphing calculator.

Use the window x [-30, 30], y [-30, 30] to capture the full parabola.

Key Takeaway. All of these properties — especially the vertex and maximum value — come straight out of the equation when it is in vertex form. No calculator needed.

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