Question 52·Easy·Equivalent Expressions

Think about the greatest number that divides both $4$ and $12$, and also notice that both terms contain $y$. Combine these into a single greatest common factor.

Once you decide on the greatest common factor, divide each original term by it. The results of those divisions are exactly what go inside the parentheses.

In Desmos, type f(x) = 4x^2 - 12x. (Using $x$ instead of $y$ is fine because the variable name does not change the relationship.)

On new lines, type:

• g(x) = 4(x - 3)
• h(x) = 2x(2x - 3)
• p(x) = 4x^2(x - 3)
• q(x) = 4x(x - 3) These correspond to the four answer choices, with $y$ replaced by $x$.

Look at the graphs of $f(x)$ and each of the other functions. The correct choice is the one whose graph lies exactly on top of the graph of $f(x)$ for all visible $x$-values (they should be indistinguishable and match at every point in the table if you create one).

Look at the two terms: $4y^2$ and $-12y$.

• The number parts are $4$ and $12$, whose greatest common factor is $4$.
• Both terms contain the variable $y$, and the smallest power of $y$ present in both is $y^1$. So the greatest common factor of the entire expression is $4y$.

Now divide each term by $4y$ to see what will be inside the parentheses:

• $4y^2 \div 4y = y$
• $-12y \div 4y = -3$ So, after factoring out $4y$, the expression becomes $4y(\text{something})$ where the "something" is $y - 3$.

Putting the GCF and the terms inside the parentheses together gives the fully factored form:

Check by distributing: $4y \cdot y = 4y^2$ and $4y \cdot (-3) = -12y$, which matches the original expression. So the equivalent expression is $4y(y - 3)$.