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How do you find the restricted values for x or the rational expression $\dfrac{{{x^2} + x + 15}}{{{x^2} - 3x}}$?

Last updated date: 12th Jul 2024
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Hint: In the above question, the concept is based on the concept of excluded values for rational for rational expressions. The main approach towards solving this expression is that we need to restrict any value for any variable in the denominator that would make that value of the denominator as zero.

Complete step by step solution:
The above given expression is an algebraic expression with numerator and denominator having the expression.
Generally, in the rational expression to simplify it we need to know that,
\[b \ne 0,\dfrac{{ab}}{b} = a\] where denominator should not be zero.
But when we need to restrict values or exclude values then it is also called as points of discontinuity.
Now these excluded values that make denominators equal to zero are not a part of the denominator.
Here the above given expression is $\dfrac{{{x^2} + x + 15}}{{{x^2} - 3x}}$.We need to look at the expression at the denominator and equate it with zero since we need to find the excluded values
\[{x^2} - 3x = 0\]
Now we can take x common from the expression we get,
x\left( {x - 3} \right) = 0 \\
x = 0 \\
\[x = 3\]
Hence, we get the above two values 0 and 3 and these values are already excluded from the domain of the rational expression.

Note: An important thing to note is that a value that makes the rational expression in the lowest form undefined then it is called an excluded value. Since we are not allowed to divide by zero, so these values are important to identify and exclude while solving.