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

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

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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?

There are lots of ways of approaching this problem. One way would be to set both of the expressions or functions equal to each other. This gives us 𝑥 squared plus 12𝑥 plus 36 over 𝑥 squared minus 𝑎 is equal to 𝑥 plus six over 𝑥 minus six. The numerator on the left-hand side is a quadratic that can be factorized into two parentheses or brackets. As the leading term has a coefficient of one, the first term in each of our brackets will be 𝑥. We need to find a pair of numbers that have a product of 36 and a sum of 12.

There are five factor pairs of 36: one and 36, two and 18, three and 12, four and nine, and six and six. The only pair that sum to 12 is six and six. The factorized form of 𝑥 squared plus 12𝑥 plus 36 is therefore equal to 𝑥 plus six multiplied by 𝑥 plus six. Our next step is to divide both sides of the equation by 𝑥 plus six. This gives us 𝑥 plus six over 𝑥 squared minus 𝑎 is equal to one over 𝑥 minus six. We can then cross multiply. We can multiply both sides of the equation by 𝑥 squared minus 𝑎 and 𝑥 minus six. This gives us 𝑥 plus six multiplied by 𝑥 minus six on the left-hand side and 𝑥 squared minus 𝑎 on the right-hand side.

We could expand the left-hand side using the FOIL method, multiplying the first, outside, inside, and last terms. This would give us 𝑥 squared minus six 𝑥 plus six 𝑥 minus 36. The negative six 𝑥 and positive six 𝑥 here cancel. Alternatively, we might have noticed that 𝑥 plus six multiplied by 𝑥 minus six was the difference of two squares and was therefore equal to 𝑥 squared minus 36. We know that this is equal to 𝑥 squared minus 𝑎. Our value of 𝑎 is therefore equal to 36.

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