Monthly Archives: May 2018

Equations, expressions, and functions

Three things that students of algebra often confuse are the notions of equations, expressions, and functions.


An expression is any set of constant numeric values, variables, and operations. The idea is that, if we know the values of all the variables at a given point, we can determine the numeric value of the entire expression.

Some expressions don’t have any variables at all. This usually means that they have a specific, unchanging mathematical value. The only exception to this is when some portion of the expression breaks math. Dividing by zero, for instance, always breaks math, so \(5/0\) has no mathematical value. Taking square roots of negative numbers leads to mathematical values, but not real ones, so depending on our goal, \(\sqrt(-5)\) may not have a valid value.

Since some values can create problems, this means that some expressions with variables have specific values or sets of values that create problems. For instance, if we’re limiting ourselves to real numbers, \(\sqrt(x)\) prohibits \(x\) from being negative. And \(1/x\) prohibits \(x\) from being zero.


An equation is a statement of fact about two expressions. An equation is true for any values of variables that make the two expressions have the same mathematical value, and otherwise it is false.

For instance, consider \(x^2 = 4\). This is true for any numbers which, when squared, have a value of 4. These are 2 and -2. Those are the only values that make that statement true.

We can have variables on both sides of the equation. Consider \(x^2 = x + 2\). This is also true for two values, 2 and -1.

If an equation involves two or more variables, it can be true for an infinite number of values. For instance, \(y = x\) is true for all pairs \((x, y)\) where \(x\) and \(y\) have the same value.


A function is a relationship between sets of data. It is often described as being a machine: If you put a specific value into the machine, you can predict exactly what output you’ll get.

It can very often by described by an expression, but it is rarely described by an equation. This is confusing because we normally state a function by giving an equation. But let’s take a careful look at a function. Here’s an example: \[f(x) = x^2 + 7\]

This is an equation, but it’s connecting two expressions. One expression consists of the name of a function (\(f\)) and its input (\(x\)). The other expression describes what operations are going to be applied to the input. The expression on the right is the actual function; the expression on the left gives its name.

Consider this: “My car is a silver Honda.” What is my car? It’s not the entire sentence “My car is a silver Honda.” That would be silly. It’s the object that both “my car” and “a silver Honda” refer to. In a similar way, \(f(x)\) and \(x^2 + 7\) are two ways to refer to a machine that takes any value, squares it, and adds seven to the result.

We use \(f(x)\) when we want to make general comments, or when we want to make other statements about the function. We use the specific form of the function (\(x^2 + 7\)) when we want to see what it actually does.

Factoring Quadratics

Today in class I presented a technique for factoring quadratics. Here are the steps. Remember that we’re starting from the right. This only works when the coefficient on the \(x^2\) term is 1.

  1. List the pairs of numbers that multiply to the constant (ignoring the sign!).
  2. The sign on the constant tells us whether we’re looking for a sum or a difference.
  3. The coefficient on the \(x\) term (ignoring the sign!) tells us what sum or difference we’re looking for.
  4. Now, using the sign on the \(x\) term, decide the signs for the factors.
    1. The higher factor will use the same sign.
    2. The lower factor will use the same sign if it’s a sum, and the opposite sign if it’s a difference.

Here are some examples.

\[x^2 + 7x + 12\]

  1. \(12 = 1 \times 12 = 2 \times 6 = 3 \times 4\)
  2. We’re looking for a sum. Our options are:
    1. \(1 + 12 = 13\)
    2. \(2 + 6 = 8\)
    3. \(3 + 4 = 7\)
  3. Our sum is 7. That means we want to use 3 and 4.
  4. We’re using +. So our factors are:
    1. \(x + 4\)
    2. \(x + 3\)

That means that \(x^2 + 7x + 12 = (x + 3)(x + 4)\).

\[x^2 – 2x – 63\]

  1. \(63 = 1 \times 63 = 3 \times 21 = 7 \times 9\)
  2. We’re looking for a difference. Our options are:
    1. \(63 – 1 = 62\)
    2. \(21 – 3 = 18\)
    3. \(9 – 7 = 2\)
  3. Our difference is 2. That means we want to use 7 and 9.
  4. We’re using -. So our factors are:
    1. \(x – 9\)
    2. \(x + 7\)

That means that \(x^2 – 2x – 63 = (x – 9)(x + 7)\).

Remember that the solutions will have the opposite signs to the factors, because solutions are values that make the factors equal to zero. So, in our first example, our solutions are \(x = {-3, -4}\), while in the second example, they’re \(x = {9, -7}\).

If this method doesn’t work, it means that at least one of the solutions of the quadratic expression isn’t an integer.