Completing The Square

This assumes:

This is related to:

This is a technique you can use to reduce the number of times a variable appears in a expression. In this case the variable x in the expression $x^2 + bx$:

Solution

$x^2 + bx = (x + \frac{b}{2})^2 - (\frac{b}{2})^2$

Geometrical proof

  1. Imagine the square $x^2$ and the rectangle bx.
  2. Split bx in two at b, preserving the x-side so you are left with $bx = \frac{b}{2}x + \frac{b}{2}x$
  3. Attach the new rectangles by either of their x-sides to two unique sides of the square that are not parallel to one another.
  4. This almost forms a new square as what is missing is the small square $(\frac{b}{2})^2$ in one of the corners (depending on how you arranged them).
  5. To “complete the square” we could add the final piece $(\frac{b}{2})^2$ to the mix. But, instead of adding it and getting a completely new square with a irrelevant area, we can subtract it from this new hypothetical square and remain in balance with our original expression.
  6. Keeping this in mind, the area of our new hypothetical square has to be $(x + \frac{b}{2})^2$. If we now subtract $(\frac{b}{2})^2$ we have the same area as our original non-square shape $x^2 + bx$
  7. This means: $x^2 + bx = (x + \frac{b}{2})^2 - (\frac{b}{2})^2$
  8. Now we have a equivalent expression with x present only one time.

Algebraic Sum-up

If we want to know x in $x^2 + bx$, we need to write it a different way:

  1. $x^2 + bx$ # The original expression
  2. $x^2 + bx = x^2 + \frac{b}{2}x + \frac{b}{2}x$ # Splitting bx rectangle
  3. $x^2 + bx + (\frac{b}{2})^2 = x^2 + \frac{b}{2}x + \frac{b}{2}x + (\frac{b}{2})^2$ # “Completing the Square”
  4. $x^2 + bx + (\frac{b}{2})^2 = (x + \frac{b}{2})^2$ # Performing the geometry hack
  5. $x^2 + bx = (x + \frac{b}{2})^2 - (\frac{b}{2})^2$ # Back to original square (with one x)