Solving Equations of the Form $\,xy = 0\,$

A very important type of equation is one of the form ‘$\,xy = 0\,$’. This type of equation has zero on one side (usually the right-hand side), and things being multiplied on the other side.

How can a sentence of this form be true? To answer this question, consider the following. Suppose I were to say to you:

I'm thinking of two numbers. When I multiply these numbers together, I get zero.

Can you tell me anything about the numbers I'm thinking of? Indeed! The only way that numbers can multiply to give zero is if at least one of the numbers is zero:

$3\cdot 0 = 0$ $0\cdot\frac12 = 0$ $0\cdot 0 = 0$ $(3.7)\cdot 0 = 0$ and so on $\ldots$

That is, in order for the sentence ‘$\,xy = 0\,$’ to be true, either $\,x\,$ must equal $\,0\,,$ or $\,y\,$ must equal $\,0\,,$ or both must equal zero.

The ‘official’ statement of this property is called the Zero Factor Law. Don't worry right now if the formal statement below doesn't make sense to you. You'll learn how to ‘translate’ it in upcoming sections.

the Zero Factor Law

Let $\,x\,$ and $\,y\,$ be real numbers. Then: $$ \cssId{s30}{xy = 0}\ \ \ \ \cssId{s31}{\text{is equivalent to}}\ \ \ \ \cssId{s32}{(x = 0\ \text{ or }\ y = 0)} $$

With this idea in mind, consider the equation:

$\,x(x-1) = 0\,$

The things being multiplied on the left-hand side are:

$\,x\,$ and $\,x-1\,$

In order for the equation to be true, either:

$\,x = 0\,$ or $\,x-1 = 0\,$

Consequently, the only numbers that make the equation true are $\,0\,$ and $\,1\,.$

Note

When presented with an equation like ‘$\,x(x-1) = 0\,$’ or ‘$\,(x-2)(x+3) = 0\,$’, many people's first reaction is to start multiplying out the left-hand side. RESIST THIS TEMPTATION! Multiplying out does absolutely no good (except perhaps giving practice with multiplying polynomials).

The correct thought process is: Ah hah! I have things being MULTIPLIED that equal ZERO. So... one of the things being multiplied must equal zero!

Example

Question:
Determine the values of $\,x\,$ for which the sentence is true: $(x-1)(x+3) = 0$

Solution:
What makes $\,x-1\,$ equal to zero? Answer: $\,1$

What makes $\,x + 3\,$ equal to zero? Answer: $\,-3$

Thus, the values of $\,x\,$ for which the sentence is true are: $\,1, -3$

Practice

In the web exercise, you must:

Put the solutions in the order you get them from left to right in the equation;
Put a comma between each number;
List repeated answers each time they occur.

For example, for this equation:

$$(x-1)(x-1)(x+2) = 0$$

You would input the answer: 1,1,-2

For more advanced students, a graph is available: e.g., the equation $\,(x-1)(x-1)(x+2) = 0\,$ is optionally accompanied by the graph of $\,y = (x-1)(x-1)(x+2)\,.$ Notice that you are finding the place(s) where this graph crosses the $x$-axis. Click the ‘Show/Hide Graph’ button to toggle the graph.