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

Which of the following is a perfect square? [A] π₯Β² β 81 [B] π₯Β² + 81 [C] π₯Β² β 18π₯ + 81 [D] π₯Β² β 18π₯ β 81 [E] π₯Β² β 9π₯ + 81

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### 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π₯ plus π all squared is equal to π squared π₯ squared plus two πππ₯ plus π squared. We can note that the expanded form of the perfect square has a constant term of positive π 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 π that will give a constant term of negative 81.

In the expanded form, we can note that the coefficient of π₯ squared is π squared, and the constant term is π squared. Our remaining three options all have the same constant term, 81, and the same leading coefficient, one. We can solve π squared equals one to obtain that π is one or negative one, and the equation π squared equals 81 to get that π equals nine or negative nine.

We can now consider the possible coefficients of π₯ by considering the possible values of two ππ. Since π is either one or negative one and π is either nine or negative nine, we can calculate that two ππ is equal to either positive or negative 18 depending on whether the signs of π and π 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 and π must be different.

Now, if we set π equals one and π 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 equals negative one and π 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.