Question 9·Hard·Advanced Math

You are told that $q = a - 3b$. Look at the denominator $a - 3b + z$. How can you rewrite that using $q$?

Consider $a^2 - 6ab + 9b^2$. Can you factor this part as a square of a binomial? How does that relate to $a - 3b$ and $q$?

Once you rewrite the numerator in terms of $q$ and $z$, look for the pattern $A^2 - B^2 = (A - B)(A + B)$. After factoring, see if any factor in the numerator matches the denominator so that the fraction can be simplified.

In Desmos, create sliders for $a$, $b$, and $z$ (for example, from $-5$ to $5$). Then enter the expression

Make sure to avoid slider values where the denominator $a - 3b + z$ equals $0$.

Enter $q = a - 3b$ as a separate line. Then enter four more lines for the answer choices:

• $A = q + z$
• $B = (q - z)^2$
• $C = (q + z)(q - z)$
• $D = q - z$. Desmos will show numerical values for $E$, $A$, $B$, $C$, and $D$ for each set of slider positions.

Move the sliders for $a$, $b$, and $z$ to several different non-problematic values (where the denominator is not $0$). For each set of values, compare the value of $E$ to $A$, $B$, $C$, and $D$. The correct choice is the one whose value always matches $E$ for every slider setting you try.