Question 137·200 Super-Hard SAT Math Questions·Advanced Math

Because $pq=756$, $p$ isn't zero, so this is a quadratic. For $ax^2+bx+c=0$, exactly one real solution means $b^2-4ac=0$: the square root in the quadratic formula is zero, making its two versions agree. Here $a=3p$, $b=-7q$, and $c=12p$. Substitute them, keeping the minus sign on $b$:

Type $4\cdot3\cdot12$ in Desmos; it shows $144$. Square $-7q$ and use that result to simplify the discriminant equation:

Add $144p^2$ to both sides to isolate the squared terms:

Take square roots, keeping absolute values because either coefficient could be negative:

The given $pq=756$ is positive, so $p$ and $q$ have the same sign. Their absolute values have a $7$-to-$12$ relationship, so:

Let $t=p/7$. Write both coefficients using that shared number:

Zero discriminant gives a ratio; the product will set its size.

Substitute $p=7t$ and $q=12t$ into the given product:

Multiply the factors:

Let $x_1=|t|$, the positive size of $t$. Type $84x_1^2\sim756\{x_1>0\}$ in Desmos. The $\sim$ asks Desmos to fit $x_1$, and the restriction keeps the positive size. Under PARAMETERS, it shows $x_1=3$.

Add the expressions for $p$ and $q$, then take the absolute value:

Combine the terms inside the absolute value:

Take the absolute value of the positive factor $19$:

Type $19x_1$ below the regression; Desmos shows $57$. So $|p+q|=57$. Choice A.