Question Video: Identifying Perfect Square Trinomials Mathematics • 9th Grade

Which of the following is a perfect square? [A] π‘₯Β² βˆ’ 81 [B] π‘₯Β² + 81 [C] π‘₯Β² βˆ’ 18π‘₯ + 81 [D] π‘₯Β² βˆ’ 18π‘₯ βˆ’ 81 [E] π‘₯Β² βˆ’ 9π‘₯ + 81


Video Transcript

Which of the following is a perfect square? Option (A) π‘₯ squared minus 81. Option (B) π‘₯ squared plus 81. Option (C) π‘₯ squared minus 18π‘₯ plus 81. Option (D) π‘₯ squared minus 18π‘₯ minus 81. Or is it option (E) π‘₯ squared minus nine π‘₯ 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 π‘Žπ‘₯ plus 𝑏 all squared is equal to π‘Ž squared π‘₯ squared plus two π‘Žπ‘π‘₯ 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 π‘₯ squared is π‘Ž 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 π‘Ž squared equals one to obtain that π‘Ž 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 π‘₯ by considering the possible values of two π‘Žπ‘. Since π‘Ž is either one or negative one and 𝑏 is either nine or negative nine, we can calculate that two π‘Žπ‘ is equal to either positive or negative 18 depending on whether the signs of π‘Ž and 𝑏 match. Only option (C) has this as a possibility, and we see that it has negative 18 as the coefficient of π‘₯, so the signs of π‘Ž and 𝑏 must be different.

Now, if we set π‘Ž equals one and 𝑏 equals negative nine, we can calculate that π‘₯ minus nine all squared is π‘₯ squared minus 18π‘₯ plus 81, which matches option (C). Interestingly, we can also choose π‘Ž equals negative one and 𝑏 equals nine to get that negative π‘₯ plus nine all squared equals π‘₯ squared minus 18π‘₯ 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.

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