Today’s guest tutorial comes to you courtesy of X.


Space

(NOTES:Tiling)

(NOTES:Origin, position)

(NOTES:Maybe could use this to discuss lines?)

Area

(NOTES:Bring in stuff from triangles article)

(NOTES:Fractional area)

(NOTES:Multiplication, division)

(NOTES:Square, root)

Distribution

(NOTES:Adding up sub-areas)

(NOTES:(X + A + B)(Y + C), etc. as decomposition)

Less is more

(NOTES:Similar, but subtracting sub-areas from larger area)

(NOTES:Add (X - A)(Y - B) as side note)

Combining terms

(NOTES:Adding like terms: 1, x, x^2)

(NOTES:Combining equations, eliminating)

The quadratic formula

(NOTES:Uses stuff from last few sections)

(NOTES:Code, along with some robustness improvements)

We come to interesting problems with $a x^2 + b x .$

We can get a hint of how to approach this by putting a $a x \times x$ rectangle next to a $b \times x$ one.

However, that $x^2$ bit suggests a square root. In that case, we can begin by resizing the first bit to a square with $\sqrt{a}{x}$ per side.

To accommodate this we also want to scale our other rect to have the same height. In doing so, we will adjust its width to preserve its size, namely to $\frac{b}{\sqrt{a}}.$

However, now our overall object is non-square. We can address this by cutting our adjusted second rect in half and putting the second piece, rotated, beneath the original square. The respective dimensions are now $\frac{b}{2\sqrt{a}}$ each.

This will still leave a gap in the corner. We can "complete the square" by taking this amount, $\frac{b^2}{4a},$ into account.

Altogether we can say $a x^2 + b x + \frac{b^2}{4a} = T,$ where $T$ is the total area of the square.

With a little massaging, we can see this has been factored out: \begin{align} a x^2 + b x + \frac{b^2}{4a} & = a\left(x^2 + \frac{b x}{a}\right) + a\frac{b^2}{4a^2} \\ & = a\left(x^2 + \frac{b x}{a} + \frac{b^2}{4a^2}\right) \\ & = a\left(x + \frac{b}{2a}\right)^2 \end{align} If we put this back together we then have: \begin{align} a\left(x + \frac{b}{2a}\right)^2 &= T \\ \left(x + \frac{b}{2a}\right)^2 &= \frac{T}{a} \end{align} This is amenable to our new square root tool: \begin{align} x + \frac{b}{2a} &= \pm \sqrt{\frac{T}{a}} \\ x &= -\frac{b}{2a} \pm \sqrt{\frac{T}{a}} \end{align} We might want to express this in terms of our known area, say $A:$ \begin{align} x &= \frac{-b}{2a} \pm \sqrt{\frac{1}{a}\left(A + \frac{b^2}{4a}\right)} \\ &= \frac{-b}{2a} \pm \sqrt{\frac{4aA + b^2}{4a^2}} \\ &= \frac{-b \pm \sqrt{4aA + b^2}}{2a} \end{align} Now, although this was tailored toward problems like $a x^2 + b x = A,$ we have something more general and can really deal with equations like $a x^2 + b x + c = 0.$ Making the substitution $c = -A,$ we have: \begin{align} x &= \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \end{align} This is the quadratic equation.

Opening lines

Our first order of business is to say a little about lines.

Lines are one-dimensional objects, meaning if we want to travel along one, we only have one direction available to follow, either forward or backward. If we veer to the side, we leave the line.

Many times, we want to consider lines in connection with their surroundings, perhaps as part of another object. By doing so, we embed them in space (or "in the plane", in two dimensions). This introduces position and orientation.

(NOTES: Expand on that last bit; some of what's in "Going the distance" would be better off here)

Curves are another sort of one-dimensional object, where the local direction changes from position to position. They look like line segments up close. Lines are particular in that by going from point to point our relative positions are the same in the space itself.

Lines will go on forever. But little bits of it are important. We can bound in one direction and we get rays. Or from both sides and then we have segments.

(NOTES: The last couple paragraphs are probably the ideas we want, but written quickly and need polish.)

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