Question 10·Easy·Equivalent Expressions

After you factor out $y$, make sure the remaining expression inside the parentheses looks like $9 - (\text{something})y$ so you can compare it directly to $9 - py$.

Once your expression is in the form $y(9 - (\text{number})y)$, compare the coefficient of $y$ inside the parentheses with the $p$ in $y(9 - py)$.

In Desmos, type f(y) = 9y - 4y^2 to define the original expression as a function of $y$.

Type g(y) = y(9 - p*y) and let Desmos create a slider for p. This represents the general form $y(9 - py)$.

Move the slider for $p$ until the graphs of $f(y)$ and $g(y)$ lie exactly on top of each other for all $y$. The value of $p$ at that point is the answer you need.

Both terms in $9y - 4y^2$ contain a factor of $y$.

Factor out $y$:

Now the expression is written in a factored form with $y$ outside the parentheses.

The problem says the expression can be written as $y(9 - py)$.

Your factored form is $y(9 - 4y)$.

Compare the inside of the parentheses:

• Given pattern: $9 - py$
• Your expression: $9 - 4y$

The structure $9 - (\text{number})y$ matches in both.

From the comparison:

• In the pattern $y(9 - py)$, the coefficient of $y$ inside the parentheses is $p$.
• In your factored form $y(9 - 4y)$, the coefficient of $y$ inside the parentheses is $4$.

So $p = 4$, which corresponds to answer choice D.