Question Video: Finding the Value of an Unknown in a Rational Function given Its Simplified Form | Nagwa Question Video: Finding the Value of an Unknown in a Rational Function given Its Simplified Form | Nagwa

# Question Video: Finding the Value of an Unknown in a Rational Function given Its Simplified Form Mathematics • Third Year of Preparatory School

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Given that 𝑛(𝑥) = (𝑥² + 12𝑥 + 36)/(𝑥² − 𝑎) simplifies to 𝑛(𝑥) = (𝑥 + 6)/(𝑥 − 6), what is the value of 𝑎?

03:15

### Video Transcript

Given that 𝑛 of 𝑥 is equal to 𝑥 squared plus 12𝑥 plus 36 over 𝑥 squared minus 𝑎 simplifies to 𝑛 of 𝑥 is equal to 𝑥 plus six over 𝑥 minus six, what is the value of 𝑎?

We begin by recalling that to simplify a rational function, we find its domain, factor the numerator and denominator, and then cancel the shared factors over the domain. In this question, the simplified function has a linear denominator, whereas the original function has a quadratic denominator. We must therefore be able to factor 𝑥 squared minus 𝑎 into two linear factors, and one of these must be 𝑥 minus six. We notice that our original expression appears to be of the form 𝑥 squared minus 𝑦 squared. And this can be factored using the difference of two squares into two linear factors: 𝑥 minus 𝑦, 𝑥 plus 𝑦. Since 𝑦 is equal to six, we have 𝑥 squared minus 𝑎 is equal to 𝑥 minus six multiplied by 𝑥 plus six. And distributing the parentheses or noting that 𝑎 is equal to 𝑦 squared, the value of 𝑎 is equal to 36.

We can verify this by considering the initial expression for 𝑛 of 𝑥. If 𝑎 is equal to 36, 𝑛 of 𝑥 is equal to 𝑥 squared plus 12𝑥 plus 36 over 𝑥 squared minus 36. The numerator can be factored into two sets of parentheses, where the first term is 𝑥. The second terms in each of the parentheses must have a product of 36 and a sum of 12. 𝑥 squared plus 12𝑥 plus 36 is equal to 𝑥 plus six multiplied by 𝑥 plus six or 𝑥 plus six all squared. As we have already factored the denominator, 𝑛 of 𝑥 is equal to 𝑥 plus six multiplied by 𝑥 plus six over 𝑥 minus six multiplied by 𝑥 plus six. The denominator has zeros at 𝑥 equals positive and negative six. Therefore, the function is undefined at these values. This means that the domain of 𝑛 of 𝑥 is the set of all real values minus the set containing negative six and six. As negative six is not contained in the domain, we can cancel the factor of 𝑥 plus six. And the function 𝑛 of 𝑥 simplifies to 𝑥 plus six over 𝑥 minus six as required. This confirms that the value of 𝑎 is 36.

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