Which of the following is a perfect square? Option (A) 𝑥 squared minus 81. Option (B) 𝑥 squared plus 81. Option (C) 𝑥 squared minus 18𝑥 plus 81. Option (D) 𝑥 squared minus 18𝑥 minus 81. Or is it option (E) 𝑥 squared minus nine 𝑥 plus 81?
In this question, we need to determine which of five given expressions is a perfect square. We can see that the five expressions we are given are quadratic polynomials in one variable and they are all written in standard form.
We can recall that a perfect square means that the expression factors to give a square expression. We also know that factoring quadratics of this form will give linear binomial factors. Thus, we are looking for an expression of the form 𝑎𝑥 plus 𝑏 all squared is equal to 𝑎 squared 𝑥 squared plus two 𝑎𝑏𝑥 plus 𝑏 squared. We can note that the expanded form of the perfect square has a constant term of positive 𝑏 squared. This is always nonnegative. We can use this to eliminate any option that has a negative constant term, such as options (A) and (D), since there is no possible value of 𝑏 that will give a constant term of negative 81.
In the expanded form, we can note that the coefficient of 𝑥 squared is 𝑎 squared, and the constant term is 𝑏 squared. Our remaining three options all have the same constant term, 81, and the same leading coefficient, one. We can solve 𝑎 squared equals one to obtain that 𝑎 is one or negative one, and the equation 𝑏 squared equals 81 to get that 𝑏 equals nine or negative nine.
We can now consider the possible coefficients of 𝑥 by considering the possible values of two 𝑎𝑏. Since 𝑎 is either one or negative one and 𝑏 is either nine or negative nine, we can calculate that two 𝑎𝑏 is equal to either positive or negative 18 depending on whether the signs of 𝑎 and 𝑏 match. Only option (C) has this as a possibility, and we see that it has negative 18 as the coefficient of 𝑥, so the signs of 𝑎 and 𝑏 must be different.
Now, if we set 𝑎 equals one and 𝑏 equals negative nine, we can calculate that 𝑥 minus nine all squared is 𝑥 squared minus 18𝑥 plus 81, which matches option (C). Interestingly, we can also choose 𝑎 equals negative one and 𝑏 equals nine to get that negative 𝑥 plus nine all squared equals 𝑥 squared minus 18𝑥 plus 81. This shows that there are two possible ways to factor a perfect square binomial in the same way that we can factor a perfect square number in different ways.