The Astounding Power of Area

[This material appears in Section 2 of the full course on Quadratics.]

In a beginning algebra course one studies linear equations, that is equations that involve expressions of the form $$ax+b$$ with $$x$$ the variable and $$a$$ and $$b$$ specific numbers. A natural next question is to wonder about expressions and equations that involve $$x^{2}$$.

JargonAn expression of the form $$ax^{2} + bx+ c$$ with $$x$$ the variable and $$a$$, $$b$$ , and $$c$$ fixed values (with $$a\ne 0$$) is called quadratic. To solve a quadratic equation means to solve an equation that can be written in the form $$ax^{2} + bx+ c = d$$.

THE NAME IS WEIRD!

The prefix quad means “four” and quadratic expressions are ones that involve powers of $$x$$ up to the second power, not the fourth power. So why are quadratic equations associated with the number four? Shouldn’t we have a name that is about the number two: Diatics? Duo-atics? Bi-atics? Maybe two-datics?

Small Aside: Have you noticed that many words in the English associated with the number two begin with “tw”? As examples: twin, twine (two threads intertwined), twixt, twilight, and the number two itself!

SO … WHY THE NAME QUADRATIC?

Quadratic equations are intimately connected with problems about squares and quadrangles (another name for rectangles). In fact, the word quadratic is derived from the Latin word quadratus for square. For example, consider the problem:

A quadrangle has one side four units longer than the other. Its area is 60 square units. What are the dimensions of the quadrangle?

If we denote the length of one side of the quadrangle as $$x$$ units, then the other must be $$x + 4$$ units in length. We have the equation $$x(x+4)=60$$ to contend with, which is equivalent to the quadratic equation $$x^{2}+4x=60$$.

But there’s more to it.

As we shall soon see, solving a quadratic equation, even if it does not come from a quadrangle problem, still involves making use of the geometry of a four-sided shape, in particular, the geometry of a square. It is really for this reason that equations of this form became known as the ones “solvable by a quadrangle method.”

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