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Completing the Square

Completing the square is a process that helps us rewrite a quadratic that's in the form \(ax^2+bx+c\) into the form \(a(x -  h)^2+k\). So, why would we want to do this? Well, for starters, if you want to graph the parabola, \(a(x -  h)^2+k\) is a form that's easier to work with because you automatically know that the vertex is at \((h, k)\).

(Also, if you're taking calculus, then knowing how to complete the square may help you integrate a complicated function; or, if you're taking a course in differential equations, it may help you to find the transformation of a function).

How to Complete the Square

Starting with \(ax^2+bx+c\), here's how to complete the square:

  1. Factor \(a\) (the coefficient of \(x^2\)) out of the first two terms only:
    \[
    a\left(x^2 + \frac{b}{a}x\right) + c
    \]

  2. In your head, divide the coefficient of \(x\) by 2, square it, and whatever you get, add and subtract that inside the brackets (since you're adding AND then subtracting, nothing changes!)\[
    a\left(x^2 + \frac{b}{a}x + \left(\frac{b}{2a}\right)^2 - \left(\frac{b}{2a}\right)^2\right) + c
    \]
  3. Bring the new term you're subtracting outside of the brackets, but remember that it gets multiplied by \(a\) when you do so.\[
    a\left(x^2 + \frac{b}{a}x + \left(\frac{b}{2a}\right)^2\right) + c - a\left(\frac{b}{2a}\right)^2
    \]
  4. The terms inside the brackets are a perfect square!

\[a\left(x + \frac{b}{2a}\right)^2 + c - a\left(\frac{b}{2a}\right)^2\]


OK, it's quite tricky to make sense of this just by looking at the steps, so hopefully the examples should make a lot more sense!

Example: Complete the square to write \(x^2 + 6x + 11\) in the form of \(a(x -  h)^2+k\).

Solution:

\(x^2 + 6x + 11\)

\(=(x^2+ 6x + 9 - 9) + 11\)

\(=(x^2 + 6x + 9) + 11 - 9\)

\(=(x^2 + 6x + 9) + 2\)

\(= (x + 3)^2+ 2\)

Example: Complete the square to write \(2x^2- 16x + 10\) in the form \(a(x -  h)^2+k\).

\(2x^2- 16x + 10\)

\(=2(x^2 - 8x) + 10\)

\(=2(x^2 - 8x + 16 - 16) + 10\)

\(=2(x^2 - 8x + 16) -32 + 10\)

\(=2(x - 4)^2-22\)

An Important Note:

Remember that if you expand the expression that you get for your final answer, you'll get the original expression in the question... this is a useful way to check your work!

Example 1:

Example 2:

Example 3: