Consider the formula 𝑁 equals 𝑛 times 𝑛 minus six plus nine, where 𝑛 is an
integer greater than zero. Robert realises that 𝑁 is always a square number and says that the minimum value of
𝑁 is one. Comment on his statement.
Before we can comment on his statement, we need to take a closer look at this
formula. Robert says that 𝑁 is always a square number. Where is that coming from?
If we expand this multiplication over the brackets, 𝑛 times 𝑛 equals 𝑛
squared. 𝑛 times negative six equals negative six 𝑛. And then bring down the plus nine. Even after we expand, it’s still not clear to us that 𝑁 is a square number. So let’s try to factorise this formula.
We need some values that when multiplied together produce positive nine and when
added together produce negative six. Negative three times negative three equals positive nine, and negative three plus
negative three equals negative six. And we can rewrite our formula 𝑁 equals 𝑛 minus three times 𝑛 minus three, which
is 𝑛 minus three squared. This is how we know that 𝑁 is always a square number.
But Robert also says that the minimum value of 𝑁 is one. However, we recognise a case where 𝑁 could be zero. If 𝑛 is three, three minus three equals zero, and zero squared equals zero. If 𝑛 equals three, then 𝑛 equals, in the original format, three times three minus
six plus nine, which equals three times negative three plus nine. Negative nine plus nine equals zero. When 𝑛 equals three, 𝑁 equals zero. And zero is less than one. Therefore, Robert’s statement is wrong. One is not the minimum value of 𝑁.
Part b) wants to know what value of 𝑛 will give us 𝑁 equal to 100.
When we factorised the formula, we found that 𝑁 equals 𝑛 minus three squared. We want to know when 𝑁 equals 100. So we set 100 equal to 𝑛 minus three squared. We can get rid of that square by taking the square root of both sides. The square root of 𝑛 minus three squared is 𝑛 minus three. And the square root of 100 is 10.
This is where a common error occurs. Don’t forget that 100 has two square roots, both positive and negative 10, because 10
times 10 equals 100, but negative 10 times negative 10 also equals 100. From here, let’s consider these two cases separately. 10 equals 𝑛 minus three, and negative 10 equals 𝑛 minus three.
First, in the positive case, we’ll add three to both sides. 10 plus three is 13 equals 𝑛. When 𝑛 is 13, 𝑁 will be equal to 100. The negative option, add three to both sides, and we get negative seven equals
𝑛. We found two solutions for 𝑛. 𝑛 equals negative seven and 𝑛 equals 13.
It’s important for us to go back and remember what has already been said about
𝑛. 𝑛 is an integer greater than zero. And that means 𝑛 equal to negative seven can’t be true for this formula. And the value of 𝑛 that makes the formula equal to 100 is 13.