Forms of the Quadratic: Terminology

Because mathematical terminology developed piecemeal over time, there are many inconsistencies which prove to be a challenge to students. One of the more obvious examples is what is called the “standard form” of the quadratic.

A quadratic equation has three common forms:

  1. \(ax^2 + bx + c = 0\)
  2. \(a(x – h)^2 + k = 0\)
  3. \(a(x – x_1)(x – x_2) = 0\)

Naming the first form

The first form mirrors the standard form of single variable polynomials, which is the sum of terms of the form \(ax^m\), where \(m \in N^0\) and the terms are in descending order based on \(m\). For example, the following are in standard form:

  • \(4x^3 + 3x – 1\)
  • \(5x^{15} – 3x^3 + x^2 + 6x\)
  • \(x^2 + 5x + 6\)

while these are not:

  • \(5 + x\) (not in order)
  • \(4x^2 – 2x^{-1}\) (not all positive powers)
  • \(6x^{2/3}\) (not all integer powers)

Therefore, it makes sense to refer to the form in (1) above as the standard form of the quadratic, since it is the standard form of the polynomial. The majority of adequately reputable sites I see online use this convention, such as Khan Academy, Math Is Fun, Math Warehouse, and Montery Institute.

This form is also sometimes called the expanded form or the general form.  This gives us three common names to choose from.

Naming the second form

The advantage of the second form is that it allows us to readily determine the vertex of the parabola represented by the quadratic, and also whether this is a maximum or minimum. Specifically, given a quadratic of the form \(a(x – h)^2 + k\), its vertex is \((h, k)\) and it opens upward if \(a\) is positive and downward if \(a\) is negative. For example, \(x^2 + 5x + 6 = (x + 2.5)^2 – 0.25\) has a vertex of \((-2.5, -0.25)\) and opens upward.

Because this form gives us the vertex, it is often called the vertex form, particularly by those sources that call the first form the standard form.

However, many sources call this form the standard form. Such sources tend to be more consistently aimed at college students, as opposed to high school students. Examples include NYU and Kent University, Ron Larson’s Precalculus with Limits, MapleSoft, and Analyze Math. In Elementary Algebra, intended for high schoolers, Ron Larson refers to the first form as the general form.

Wolfram, for its part, appears to dodge the issue. “Standard form” at MathWorld leads only to the standard form of a linear equation (i.e., \(ax + by = c\)), which is problematic in its own right, since the standard form of the single-variable polynomial equal to that linear equation is \(y = \frac{a}{b}x + \frac{c}{b}\), which is most typically called the slope-intercept form. Meanwhile, WolframAlpha doesn’t recognize “standard form of (x + 1)(x + 2)” as a meaningful request.

Naming the third form

While there doesn’t seem to be any temptation for anyone to call this the standard form, it too has at least two common names.

This form is commonly used because it allows us to quickly identify the places that the parabola crosses the \(x\)-axis. For example, \(x^2 + 5x + 6 = (x + 2)(x + 3)\) crosses the \(x\)-axis at \((0, -2)\) and \((0, -3)\).

These places have several names: Zeroes, solutions, roots, or \(x\)-intercepts. As a result of the last name, this form is sometimes called the intercept form. However, it is more commonly called the factored form.

It is also worth noting that the third form is usually shown as \(a(x + b)(x + c)\) rather than as \(a(x – x_1)(x – x_2)\). This is due to the way they’re generally solved. Indeed, there’s a valid argument to be had that \(a(x + b)(x + c)\) is the factored form and \(a(x – x_1)(x – x_2)\) is the intercept form. I’m sympathetic to this argument, but am stepping around it for the time being.

Avoiding ambiguity

One of the ostensible advantages of mathematics over natural language is its lack of ambiguity. In light of that, terminological ambiguities are best avoided as well, particularly when they’re not needed.

It seems to me that the term “standard form” is fairly firmly entrenched at the high school level for the first form, while it’s equally firmly entrenched at the college level for the second form. There are two obvious reasons for the first entrenchment:

  1. High school texts tend to avoid the vertex form, at least until Pre-Calculus. The emphasis in Algebra classes is on solving quadratics, not on graphing them, so the zeroes are more important than the vertex.
  2. By calling the expanded version the standard form, that allows for a simpler bridge in Algebra II or Pre-Calculus to the standard form of polynomials.

There may be other reasons as well. However, as I mentioned above, the phrase “standard form” may have already been used for a non-standard form of a polynomial. There are three standard forms of linear equations:

  1. \(ax + by = c\), the standard form
  2. \(y = mx + b\), the slope-intercept form
  3. \(y – y_1 = m(x – x_1)\), the point-slope form

The standard form is sometimes written as \(ax + bx + c = 0\), and is sometimes called the general form. Technically, both \(ax + by + c = 0\) and \(y = mx + b\) are in standard form of a polynomial; in the first case, it’s a two-variable polynomial, and in the second, \(y\) is the stand-in for \(f(x)\) on a single-variable polynomial. So this use of “standard form” is potentially confusing but defensible, while in the case of quadratics, it’s patently inconsistent.

As for why mathematicians at and above the college level prefer “standard form” for the vertex form, I cannot find anything specific, but I’m inclined to think it’s a combination of historical accident and the fact that it’s a crucial step in determining the quadratic formula, \[x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}\] Specifically, \(k\) is related to the discriminant: \(-4ak = b^2 – 4ac\). Indeed, one of the two common methods for converting from the expanded form to the vertex form is to use these relationships:

  1. \(h = \frac{b}{2a}\)
  2. \(k = f(h) = -\frac{b^2 – 4ac}{4a} = c – \frac{b^2}{4a}\)

The other method is to complete the square, for which the above is really just a shortcut.

Regardless of the reasons for using standard form for either of the forms it’s used for, we do have a ready set of unambiguous names:

  1. \(ax^2 + bx + c = 0\) is the expanded form
  2. \(a(x – h)^2 + k = 0\) is the vertex form
  3. \(a(x – x_1)(x – x_2) = 0\) is the factored form

As I mentioned above, there is some justification for calling the third form the intercept form rather than the factored form. However, “factored form” is apparently a much more common term at the moment, so I’ll endorse it for now.

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