Completing the Square
Transform a quadratic from standard form to vertex form by adding and subtracting (b/2)² to complete the square. Use this technique both to graph parabolas and to derive the quadratic formula.
A lesson of Algebra 1, in Quadratic Functions, about 35 minutes.
By the end of this lesson, you will be able to turn any quadratic into vertex form by supplying the missing constant, solve equations that refuse to factor, handle a leading coefficient, and see where the quadratic formula comes from.
Manufacture a Perfect Square
$$x^2 + 6x + 9 = (x + 3)^2$$
That works because $9$ is exactly the right constant. Completing the square is choosing that constant on purpose, so that a quadratic which does not factor becomes one that does.
$$\left(\frac{b}{2}\right)^2$$
Half of $b$, then squared. For $x^2 + 6x$, half of $6$ is $3$, and $3^2 = 9$. Add $9$ and you have a perfect square.
$$x^2 + 6x + \underline{9} = (x + \underline{3})^2$$
And the number inside the bracket is half of $b$, the same $3$ you already computed. Once you have halved $b$ you have both pieces, which is why this is one step rather than two.
| $x^2 + bx$ | half of $b$ | add | becomes |
|---|---|---|---|
| $x^2 + 8x$ | $4$ | $16$ | $(x+4)^2$ |
| $x^2 - 10x$ | $-5$ | $25$ | $(x-5)^2$ |
| $x^2 + 5x$ | $\tfrac{5}{2}$ | $\tfrac{25}{4}$ | $(x + \tfrac{5}{2})^2$ |
The added constant is always positive, because it is something squared. Only the number inside the bracket carries $b$'s sign.
Sort It: What Do You Add
Half of $b$, then squared.
Keeping the Equation True
You cannot simply add $9$ to an expression and carry on. Two ways to stay honest, and which one you use depends on whether you are looking at an expression or an equation.
In an equation, add it to both sides.
$$x^2 + 6x = 7$$
$$x^2 + 6x + 9 = 7 + 9$$
$$(x + 3)^2 = 16$$
In an expression, add and subtract it in the same line.
$$x^2 + 6x + 1 = x^2 + 6x + 9 - 9 + 1 = (x+3)^2 - 8$$
Net change zero, so the expression is unchanged. That second form is how you get vertex form, and it hands you the vertex $(-3, -8)$ with no formula at all.
Example 1: Solving by Completing the Square
Here is every step on one equation, worked all the way through. $$x^2 + 6x - 7 = 0$$
- Move the constant to the right. What does the equation say? x^2 + 6x = 7
- What is half of b? 3
- What do you add to both sides? 9