HELP Which expression is a perfect square?
A. x² - 4x + 4
B. x2 - 4x - 4
C. 12 - 9x + 9
D. 12 - 9x – 9

Choice A

Explanation:

A perfect square trinomial is of the form

(m+n)^2 = m^2 + 2mn + n^2

We can see why this works through use of the FOIL rule.

So the first and last terms must be perfect squares, and they must be positive. This rules out choices B and D since we have -4 and -9 as those last terms being negative.

Choice C is ruled out because the middle coefficient -9 isn't correct. The middle coefficient must be some even number since 2 is a factor of 2mn.

This leaves choice A. We can do a bit of trial and error to find that

(x-2)^2 = x^2 - 4x + 4

showing that choice A is a perfect square trinomial.

A) x² - 4x + 4

Step-by-step explanation:

Given the following special factoring formulas on perfect square trinomials:

u² + 2uv + v² = (u + v)²

u² - 2uv + v² = (u - v)²

Apply these concepts on the given set of equations to see which special factoring formulas apply on the given options. I will only work on one of the given options, and you could apply these same techniques on the other ones to compare the results with your calculations.

To find the binomial factors of quadratic equations, you must find factors with product uv, and sum v.

In Option A: x² - 4x + 4

Right off the bat, it seems like the difference of squares may apply to this equation.  You must find the factors that will product of 4 and a sum of -4.

The possible factors are:

product uv: -2 × -2  = 4,

sum v :  -2 + -2 = -4

Hence, the binomial factor of x² - 4x + 4 =  (x - 2)²

If you expand this binomial through FOIL method:

(x - 2)(x - 2) = x² - 2x - 2x + 4 = x² - 4x + 4

This means that the trinomial x² - 4x + 4 produces a perfect square binomial factors of (x - 2)².

Therefore, the correct answer is Option A) x² - 4x + 4.

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