Question 10.8.2

Imagine that we want to make a rectangular plot of land with an area of $209\,\mathrm{m^2}$, and we want it to be $8\,\mathrm{m}$ longer than it is wide. Diagram: completingsquare 1 Nowadays we would tackle this problem using the quadratic formula (see Question 7.5.38 for an example), but people in ancient times would have thought of the problem geometrically. We’re going to explore the method known as ‘completing the square’, which in essence is exactly the same idea we used to derive the quadratic formula, but this time we’ll do it without any algebra. Diagram: completingsquare 2 We can consider the plot of land as a square of unknown side length, with $8\,\mathrm{m}$ of rectangle stuck on the side.

  1. We might get the idea to shift bits of the area around to try to make a square (if we can work out the area of this square then we can work out its side length and hopefully find the dimensions of the plot of land), so we cut half of the rectangle off and stick it onto the bottom of the square: Diagram: completingsquare 3 We haven’t quite made everything into a square; there’s a piece missing in the bottom right corner. What’s the area of this missing piece? Show answer

  2. What is the total area of the big square, including the missing piece? Hence find its side length. Show hint

    Show answer

  3. Find the original dimensions of the plot of land. Show answer

  4. The steps for completing the square and solving the problem algebraically could be written out like this: \begin{align*} 209 &= x(x+8)\\ &= x^2 + 8x\\ &= (x + 4)^2 - 16\\ 225 &= (x+4)^2\\ 15 &= x+4\\ x &= 11 \end{align*} Note how each line corresponds to one of the steps we took geometrically. Show answer