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. 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. We can consider the plot of land as a square of unknown side length, with $8\,\mathrm{m}$ of rectangle stuck on the side.

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: 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

The missing piece is a square of side length $4\,\mathrm{m}$, so its area is $16\,\mathrm{m^2}$.

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

The original rectangle had an area of $209\,\mathrm{m^2}$.We’ve added $16\,\mathrm{m^2}$ onto the original $209\,\mathrm{m^2}$, making $225\,\mathrm{m^2}$. The side length is the square root of this, $15\,\mathrm{m^2}$.

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

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

$209 = x(x+8)$: this is equivalent to saying we have a rectangle of area $209\,\mathrm{m^2}$ which is $8\,\mathrm{m}$ longer than it is wide.

$x^2 + 8x$: we then consider it as a square stuck to a rectangle of width $8\,\mathrm{m}$.

$(x+4)^2 - 16$: cutting the rectangle in half and moving it to the bottom of the square results in a bigger square of side length $x+4$, but with $16\,\mathrm{m^2}$ missing.

$225 = (x+4)^2$: the total area of the square is $225\,\mathrm{m^2}$.

$15 = x+4$: hence its side length is $15\,\mathrm{m}$.

$x = 11$: therefore the width of the plot of land is $11\,\mathrm{m}$.