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SOLVING EQUATIONS OF THE FORM xy = 0
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The concepts for this exercise are summarized below.
For a complete discussion, read the text.
(Click here for solutions to the text exercises.)
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 equal to zero:
3 · 0 = 0
0 · (1/2) = 0
0 · 0 = 0
(3.7) · 0 = 0 and so on...
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.
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 .
EXAMPLES:
Determine the values of x for which each sentence is true.
(x - 1)(x + 3) = 0
Answer: 1,-3
You must put the solutions in the order you get them from left to right in the equation.
You must put a comma between each number, with no extra spaces.
List repeated answers each time they occur. For example:
(x - 1)(x - 1) = 0
Answer: 1,1