Video Transcript
By factorizing or otherwise, evaluate 97 squared plus six times 97 plus nine.
In this question, we are asked to evaluate a given expression. We could try to do this by hand. And while this would work, it is easy to make a mistake when squaring 97 and adding each of the terms together. Instead, let’s try to evaluate this expression by first factoring the given expression.
The first thing we can check is to see if all of the terms share some common factor that we can take out. In this case, 97 is a prime. And we can note that there is no nontrivial factor shared by all three terms. The next approach we can take to factor this expression is to note that there are three terms. So, we can check to see if the expression matches any form of a trinomial that we know how to factor.
We can see that the expression somewhat resembles the expanded form of a perfect square of a binomial by comparing the expression to 𝑎 plus 𝑏 all squared. We can see that our value of 𝑎 will be 97. The other two terms are not in the correct form. Since the middle term is positive, we can start by rewriting nine as three squared. This would mean that we would want 𝑏 to be equal to three. We can then see that six times 97 is equal to two times three times 97, which is two 𝑎𝑏 if 𝑎 is 97 and 𝑏 is three.
So, this expression is in the expanded form of the perfect square of a binomial. Therefore, we can factor the expression using the formula with 𝑎 equal to 97 and 𝑏 equal to three. We obtain 97 plus three all squared. We can now evaluate this expression. We have that 97 plus three is 100 and then 100 squared is 10,000.
