Question 59·SAT Desmos Regression Practice·Advanced Math

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The equation below is true for all values of $x$ , where $p$ and $q$ are real constants.

(x^2+px+q)(x^2-px+q)=x^4+2x^2+9

Which choice is the value of $p^2+q$ ?

The phrase true for all values of $x$ signals a polynomial identity: both sides agree at every input. With two unknown constants, use a Desmos identity regression to find them, then evaluate the requested expression. The factors contain $+px$ and $-px$ , so changing the sign of $p$ swaps the factors without changing their product.

Hints

Hint 1
An identity holds at every input, so the unknown constants must work at several $x$ -values. In Desmos, give $x_1$ a list of inputs and use $\sim$ to fit both constants.
Hint 2
Desmos can return either sign for $p$ . Replacing $p$ with $-p$ swaps the two factors, leaving their product unchanged. Does the expression you need depend on which sign Desmos finds?
Hint 3
Keep the fitted values available in Desmos and type the whole requested expression, $p^2+q$ . Stopping at $p$ or $q$ alone won't answer the question.

Step-by-step

Approach 1: Fit the identity in Desmos
Step 1
Find the constants
An identity holds at every $x$ , so its constants must work at several inputs. Type $x_1=[0...4]$ in Desmos, then type the given equation with $x_1$ in place of $x$ and $\sim$ in place of $=$ . The tilde asks Desmos to fit the unknown constants. Under PARAMETERS, it shows $q=3$ and $p=2$ .
Step 2
Handle either sign of p
Changing $p$ to $-p$ swaps the two factors, and changing their order doesn't change the product. So either sign of $p$ works. More importantly, both give $p^2=4$ ; you don't need to choose a sign.
Step 3
Evaluate the requested expression
Keep the regression lines and type $p^2+q$ . Desmos displays $7$ : it squares the fitted $p$ and adds $q$ . So the value of $p^2+q$ is $7$ . Choice B.

Approach 2: Match coefficients exactly
Step 1
Use the paired signs
The factors differ only in the sign of $px$ . Apply the difference of squares, $(A+B)(A-B)=A^2-B^2$ , with $A=x^2+q$ and $B=px$ :
$(x^2+px+q)(x^2-px+q)=(x^2+q)^2-(px)^2.$
Step 2
Expand the squares
Expand each square so you can see the terms to compare:
$(x^2+q)^2-(px)^2=x^4+2qx^2+q^2-p^2x^2.$
Step 3
Collect the x-squared terms
Combine the two $x^2$ terms:
$x^4+2qx^2+q^2-p^2x^2=x^4+(2q-p^2)x^2+q^2.$
Step 4
Match the coefficients
In an identity, coefficients of the same power of $x$ must match. The constant term on the right is $9$ , and its $x^2$ coefficient is $2$ , so:
$q^2=9,\qquad 2q-p^2=2.$
Step 5
Keep both possible values of q
Take the square root of $q^2=9$ . Both signs square to $9$ , so don't discard either yet:
$q=3\quad\text{or}\quad q=-3.$
Step 6
Isolate p-squared
Use $2q-p^2=2$ to get $p^2$ in terms of $q$ . Subtract $2q$ from both sides:
$-p^2=2-2q.$
Multiply both sides by $-1$ :
$p^2=2q-2.$
Step 7
Reject the impossible value of q
If $q=-3$ , the equation gives:
$p^2=2(-3)-2=-8.$
But $p$ is real, and a real number's square can't be negative. So $q=-3$ cannot work; $q=3$ .
Step 8
Find p-squared
Put the remaining value, $q=3$ , into $p^2=2q-2$ :
$p^2=2(3)-2=4.$
The square is all you need, not the sign of $p$ .
Step 9
Answer the question
Add $p^2=4$ and $q=3$ :
$p^2+q=4+3=7.$
The value of $p^2+q$ is $7$ . Choice B.

Question 59·SAT Desmos Regression Practice·Advanced Math

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