Completing the square is a method for rewriting a quadratic expression so that part of it becomes a perfect-square binomial. It helps reveal the shape and position of a parabola, especially the vertex. This is useful for graphing quadratics, solving equations, and understanding why the quadratic formula works.
The key move is adding and subtracting the same carefully chosen value so the expression stays equivalent.
Key Facts
- For x^2 + bx, add (b/2)^2 to make x^2 + bx + (b/2)^2 = (x + b/2)^2.
- Vertex form is y = a(x - h)^2 + k, where the vertex is (h, k).
- To complete the square for ax^2 + bx + c when a is not 1, first factor a from the x terms.
- x^2 + 6x + 5 = (x + 3)^2 - 4 because (6/2)^2 = 9.
- Solving by completing the square often leads to (x - h)^2 = r, so x - h = ±sqrt(r).
- For y = ax^2 + bx + c, the vertex x-coordinate is h = -b/(2a).
Vocabulary
- Quadratic expression
- An expression that can be written in the form ax^2 + bx + c, where a is not 0.
- Perfect-square trinomial
- A trinomial that factors into a binomial multiplied by itself, such as x^2 + 10x + 25 = (x + 5)^2.
- Completing the square
- A process of rewriting a quadratic by adding and subtracting a value that creates a perfect-square trinomial.
- Vertex form
- The form y = a(x - h)^2 + k, which shows the vertex of the parabola as (h, k).
- Vertex
- The highest or lowest point of a parabola, depending on whether it opens downward or upward.
Common Mistakes to Avoid
- Adding (b/2) instead of (b/2)^2 is wrong because the square term is what makes the trinomial factor into a binomial squared.
- Forgetting to subtract the same value after adding it changes the expression, so the new expression is no longer equivalent to the original.
- Ignoring the leading coefficient a when a is not 1 gives the wrong square because the x^2 coefficient must be 1 inside the completed square step.
- Reading y = a(x - h)^2 + k as having vertex (-h, k) is wrong because the sign inside the parentheses is opposite of the vertex x-coordinate.
Practice Questions
- 1 Rewrite x^2 + 8x + 3 in vertex form by completing the square.
- 2 Solve x^2 - 10x + 16 = 0 by completing the square.
- 3 Explain why x^2 + 12x can be turned into a perfect-square trinomial by adding 36, and describe how algebra tiles could show this visually.