Quadratics

2.1 Why the name “Quadratic”?

Lesson materials located below the video overview.

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. (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.”

Please join the conversation on Facebook and Twitter and kindly share this page using the buttons below.


Share on FacebookTweet about this on Twitter

Resources

resources

Books

Take your understanding to the next level with easy to understand books by James Tanton.

BROWSE BOOKSarrow

resources

Guides & Solutions

Dive deeper into key topics through detailed, easy to follow guides and solution sets.

BROWSE GUIDESarrow

light bulb

Donations

Consider supporting G'Day Math! with a donation, of any amount.

Your support is so much appreciated and enables the continued creation of great course content. Thanks!

heart

Ready to Help?

Donations can be made via PayPal and major credit cards. A PayPal account is not required. Many thanks!

DONATEarrow