Question 12·25 Super-Hard SAT Advanced Math Questions·Advanced Math

Type $x_1=[-2,-1,0,1,2]$. Then type the given equation with each $x$ changed to $x_1$ and $=$ changed to $\sim$. The $\sim$ tells Desmos to fit the unknown constants. Restrict $b$, $c$, and $d$ to be positive as the problem requires; without $b>0$, Desmos can fit a negative $b$ even when $c$ and $d$ are positive. Under PARAMETERS, the regression shows $a=3$, $b=5$, $c=4$, and $d=6$.

Type $ad+bc$ below the regression. Desmos uses the fitted constants and prints $38$. That includes both products, so $ad+bc=38$. Choice C.

Because the equation holds for every $x$, matching powers of $x$ have matching coefficients, or numbers multiplying them. Squaring $x^2+ax+b$ makes the $x^3$ coefficient $2a$; the subtracted square has no $x^3$ term. Match the cubic coefficients:

Divide by $2$:

Set $x=0$ so every term containing $x$ disappears:

Reverse both sides’ signs:

Factor this difference of squares:

The integers $b$ and $d$ are positive. Since $d^2-b^2=11$, $d-b$ is positive, and $d+b$ is larger. The only positive factor pair of $11$ is $1$ and $11$, so:

Add these equations:

Divide by $2$:

Substitute into $d+b=11$:

The $x^2$ coefficient on the left is $a^2+2b-c^2$. Match it to the $x^2$ coefficient, $3$, and substitute $a=3$ and $b=5$:

Combine the numbers:

Subtract $19$:

Multiply by $-1$:

Take the positive square root because $c$ is positive:

Type $3(6)+5(4)$ in Desmos to calculate $ad+bc$, not $ac+bd$. It prints $38$, so the requested value is $38$. Choice C.