Question 152·Medium·Equivalent Expressions

Remember $(a - b)^2 = a^2 - 2ab + b^2$ and $(a + b)(a - b) = a^2 - b^2$. Can you apply these to $(3x - 2)^2$ and $(x + 4)(x - 4)$?

After you expand both parts, you will have something like "[first polynomial] - [second polynomial]". Make sure you distribute the negative sign to every term in the second polynomial.

Once the subtraction is done, combine the $x^2$ terms, the $x$ terms, and the constant terms separately to get a single simplified polynomial.

In Desmos, type f(x) = (3x - 2)^2 - (x + 4)(x - 4) to define the original expression as a function.

Type each option as its own function, for example g(x) = 8x^2 - 12x - 20, h(x) = 10x^2 - 12x + 20, etc., so you have one graph for the original expression and one for each choice.

Look at the graphs: the correct choice will have a graph that overlaps exactly with the graph of $f(x)$ for all $x$ (they appear as a single curve).

Alternatively, create a new function like d_A(x) = f(x) - (\text{choice A expression}) for each option. For the correct choice, the graph of this difference will be a horizontal line at $y = 0$ for all $x$, showing the two expressions are equivalent.

Use the formula $(a - b)^2 = a^2 - 2ab + b^2$ with $a = 3x$ and $b = 2$.

So

Now your expression is:

Recognize $(x + 4)(x - 4)$ as a difference of squares: $(a + b)(a - b) = a^2 - b^2$.

Here $a = x$ and $b = 4$, so

Substitute this into the expression from Step 1:

Subtracting $(x^2 - 16)$ means you must change the signs of both terms inside the parentheses:

Now you have a sum of like terms to combine.

Group and combine like terms:

• $9x^2 - x^2 = 8x^2$
• The $x$ term stays $-12x$
• $4 + 16 = 20$

So the simplified expression is $8x^2 - 12x + 20$, which matches choice D.