Question 144·200 Super-Hard SAT Math Questions·Advanced Math

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If $x^2 = j + k$ and $y^2 = j + m$ , which choice is equal to $(x^2+y^2)^2-(x^2-y^2)^2$ ?

Two squared groups with a minus between them signal a difference of squares. With several letters in the choices, you can use Desmos to test one allowed set of values and eliminate mismatches. One mismatch disproves equivalence; one match alone does not prove it. Factoring the outer squares then confirms why the remaining choice works for every allowed value.

Hints

Hint 1
An equivalent expression has the same value as the original for every allowed input. One input where they differ rules a choice out. Try nonzero values for $j$ , $k$ , and $m$ so their products don't disappear.
Hint 2
The problem gives $x^2$ and $y^2$ , not $x$ and $y$ . Replace each whole square with its given expression before typing into Desmos. Then you won't need to find $x$ or $y$ .
Hint 3
Evaluate the original and every choice at the same values. A different output eliminates a choice. If three choices fail, which one remains?

Step-by-step

Approach 1: Eliminate choices with Desmos
Step 1
Pick a valid test
Type $j=1$ , $k=2$ , and $m=3$ into Desmos. These nonzero values keep the $jk$ , $jm$ , and $km$ terms visible. The given equations make $x^2=1+2=3$ and $y^2=1+3=4$ , so real values of $x$ and $y$ can satisfy this test.
Step 2
Evaluate the original expression
Replace each entire square using the given equations. Type $((j+k)+(j+m))^2-((j+k)-(j+m))^2$ below your three assignments. Desmos shows $48$ , the original expression's value at this test.
Step 3
Rule out the mismatches
Type the four complete choices below that line, in the order shown. Desmos gives $0$ , $36$ , $38$ , and $48$ . The first three cannot be equivalent because they fail this valid test. The remaining expression is $4j^2+4jk+4jm+4km$ . Choice D.

Approach 2: Prove the match by factoring
Step 1
Factor the outer squares
The difference-of-squares rule is $A^2-B^2=(A-B)(A+B)$ . Here, $A$ and $B$ are the whole expressions $x^2+y^2$ and $x^2-y^2$ . Factor them before expanding either square: $(x^2+y^2)^2-(x^2-y^2)^2=\big((x^2+y^2)-(x^2-y^2)\big)\big((x^2+y^2)+(x^2-y^2)\big)$ .
Step 2
Remove the inner brackets
Remove the inner brackets. Watch the minus before the second bracket: $\big((x^2+y^2)-(x^2-y^2)\big)\big((x^2+y^2)+(x^2-y^2)\big)=(x^2+y^2-x^2+y^2)(x^2+y^2+x^2-y^2)$ .
Step 3
Combine matching terms
Combine the matching terms in each factor: $(x^2+y^2-x^2+y^2)(x^2+y^2+x^2-y^2)=(2y^2)(2x^2)$ .
Step 4
Multiply the factors
Multiply the two factors: $(2y^2)(2x^2)=4x^2y^2$ .
Step 5
Use both given equations
The problem gives $x^2=j+k$ and $y^2=j+m$ , so replace both squares in the product: $4x^2y^2=4(j+k)(j+m)$ .
Step 6
Expand the product
Use the distributive property: each term in one parenthesis multiplies every term in the other: $4(j+k)(j+m)=4(j^2+jm+jk+km)$ .
Step 7
Distribute to match a choice
Then multiply every resulting term by $4$ : $4(j^2+jm+jk+km)=4j^2+4jm+4jk+4km$ . This matches the expression left after the Desmos test for every allowed value. Choice D.

Question 144·200 Super-Hard SAT Math Questions·Advanced Math

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