Question 61·SAT Desmos Regression Practice·Advanced Math

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For all values of $x$ , the following equation is true, where $a$ , $b$ , $c$ , and $d$ are real constants.

\begin{aligned} (x^2+ax+b)^2-(x^2+cx+d)^2 &=12x^3+10x^2-10x-21 \end{aligned}

Which choice gives the value of $a+b$ ?

The phrase for all values of $x$ signals a polynomial identity: both sides must match at every input. With several unknown constants, use a Desmos identity regression across a list of inputs, then check the fitted values with a difference of squares. Matching at one input doesn't establish an identity.

Hints

Hint 1
An identity holds at every value of $x$ . Choose several inputs, not just $x=1$ : one input gives too little information to find four unknown constants.
Hint 2
Make $x_1$ a list in Desmos. Then replace every $x$ with $x_1$ and the equals sign with $\sim$ ; a regression fits $a$ , $b$ , $c$ , and $d$ across the whole list.
Hint 3
Check that the fitted values work for every $x$ . The difference of squares rule, $U^2-V^2=(U-V)(U+V)$ , turns the two squares into a product you can compare with the given cubic.

Step-by-step

Fit the constants, then check the identity
Step 1
Give Desmos several inputs
Type $x_1=[-2...2]$ . Desmos lists $-2,-1,0,1,2$ . Because the equation holds for all values of $x$ , it must hold at each of these inputs. One input alone couldn't pin down four constants.
Step 2
Fit the four constants
Enter the given equation with every $x$ changed to $x_1$ and $=$ changed to $\sim$ . That symbol tells Desmos to fit the constants across all five inputs. Under PARAMETERS, Desmos shows $a\approx5$ , $b\approx-2$ , $c\approx-1$ , and $d\approx5$ .
Step 3
Check the fitted values for every input
A fit at five inputs isn't yet a check of every $x$ . Use the difference of squares rule, $U^2-V^2=(U-V)(U+V)$ , with the fitted integers:
$\begin{aligned} (x^2+5x-2)^2-(x^2-x+5)^2 &=(6x-7)(2x^2+4x+3)\\[1.4em] &=12x^3+10x^2-10x-21. \end{aligned}$
The exact match confirms the identity.
Step 4
Find the requested sum
Type $a+b$ below the regression. Desmos prints $3$ , using $a=5$ and $b=-2$ . The question asks for their sum, not either constant alone, so $a+b=3$ . Choice B.

Question 61·SAT Desmos Regression Practice·Advanced Math

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