Question 93·200 Super-Hard SAT Math Questions·Advanced Math

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The expression

(x^2-4x+1)^2-25(x-2)^2

can be rewritten in the form $(x^2+px+q)(x^2+rx+s)$ , where $p$ , $q$ , $r$ , and $s$ are integers and $p<r$ . What is the value of $p+s$ ?

Your answer

A difference of squares appears when one squared quantity is subtracted from another, even if a coefficient hides the second square. Rewrite both quantities as squares, use the minus/plus factor pair, and expand each factor. A Desmos graph-overlap check can catch a sign slip, but use the given ordering condition before assigning the letters.

Hints

Hint 1
Since $25=5^2$ , the entire term $25(x-2)^2$ is the square of $5(x-2)$ . Keep $x-2$ together: what two quantities are being squared in the original expression?
Hint 2
For a difference of squares, $A^2-B^2=(A-B)(A+B)$ . The two factors use the same $B$ , once with a minus sign and once with a plus sign.
Hint 3
After expanding, compare the coefficients of $x$ . The condition $p<r$ tells you which quadratic comes first. Then take $p$ from the first factor and $s$ from the second.

Step-by-step

Factor the difference of squares
Step 1
Identify the two squares
The first term is already a square. For the second, $25=5^2$ , so $25(x-2)^2=[5(x-2)]^2$ . The whole quantity $5(x-2)$ is being squared, not just the $5$ .
Step 2
Split into a minus factor and a plus factor
A difference of squares follows $A^2-B^2=(A-B)(A+B)$ . Here $A=x^2-4x+1$ and $B=5(x-2)$ , so use both signs:
$(x^2-4x+1)^2-25(x-2)^2 =[x^2-4x+1-5(x-2)][x^2-4x+1+5(x-2)].$
Step 3
Distribute in the minus factor
Distribute $-5$ through $x-2$ :
$x^2-4x+1-5(x-2)=x^2-4x+1-5x+10.$
Step 4
Combine terms in the minus factor
Combine like terms:
$x^2-4x+1-5x+10=x^2-9x+11.$
Step 5
Distribute in the plus factor
Distribute $+5$ through $x-2$ :
$x^2-4x+1+5(x-2)=x^2-4x+1+5x-10.$
Step 6
Combine terms in the plus factor
Combine like terms:
$x^2-4x+1+5x-10=x^2+x-9.$
Step 7
Check the factorization in Desmos
Type the original expression as $y=(x^2-4x+1)^2-25(x-2)^2$ , then type $y=(x^2-9x+11)(x^2+x-9)$ . The graphs overlap, which helps check the signs you used. The difference-of-squares identity is what guarantees the expressions agree for every $x$ .
Step 8
Assign the coefficients in the required order
The $x$ coefficients are $-9$ and $1$ . Since $-9<1$ , the first factor is $x^2-9x+11$ , giving $p=-9$ . The second is $x^2+x-9$ , so its constant term is $s=-9$ . Reversing the factors would break $p<r$ .
Step 9
Find the requested sum
Type $-9+(-9)$ in Desmos. It shows $-18$ , so $p+s=-18$ . Grid in -18.

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SAT Advanced AlgebraIntermediateCoreFactor algebraic expressions

Question 93·200 Super-Hard SAT Math Questions·Advanced Math

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