Mathematics
Grade10
Easy
Question
For what value of x is the expression undefined?
![fraction numerator 2 straight x squared space plus space 8 straight x over denominator left parenthesis straight x space plus space 4 right parenthesis space open parentheses straight x squared space minus space 9 close parentheses end fraction](data:image/png;base64,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)
- -8
- -3
- 0
- 4
Hint:
A rational number is a sort of real number in mathematics, and it takes the form p/q, where q is not equal to zero. Here we have given the expression
, we have to find for what value of x is the expression undefined. We can use the concept of rational numbers to find the value of x.
The correct answer is: -3
In mathematics, any number that can be written as p/q where q 0 is considered a rational number. Additionally, every fraction that has an integer denominator and numerator and a denominator that is not zero falls into the category of rational numbers. The outcome of dividing a rational number, or fraction, will be a decimal number, either a terminating decimal or a repeating decimal.
Now we have given the expression as:
![fraction numerator 2 straight x squared space plus space 8 straight x over denominator left parenthesis straight x space plus space 4 right parenthesis space open parentheses straight x squared space minus space 9 close parentheses end fraction](data:image/png;base64,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)
The expression is undefined when denominator is zero, so:
![left parenthesis x space plus space 4 right parenthesis space left parenthesis x squared space minus space 9 right parenthesis space equals space 0 space
N o w space w e space c a n space s i m p l i f y space t h e space s e c o n d space b r a c k e t space b y space u sin g space t h e space f o r m u l a colon space
a squared minus b squared equals left parenthesis a plus b right parenthesis left parenthesis a minus b right parenthesis space
U sin g space t h i s comma space w e space g e t colon
left parenthesis x space plus space 4 right parenthesis space left parenthesis x space minus space 3 right parenthesis space left parenthesis x space plus space 3 right parenthesis space equals space 0 space space
x space equals space minus 4 space comma space 3 space comma space minus 3](data:image/png;base64,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So at the values -4, -3 and 3, x is undefined.
So here the concept of rational numbers is used. When the denominator is 0, a rational expression has no specified meaning. Set the denominator to 0 and solve for x to get the values of x that lead to a denominator of 0. So at the values -4, -3 and 3, x is undefined.