If and , which choice is equal to ?

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Super-Hard SAT Math Question 144 | Free SAT Prep | Aniko

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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 $(x^2+y^2)^2-(x^2-y^2)^2$ ? аnікο.aі/‌sаt

When you see two squared expressions being subtracted, look immediately for a difference of squares structure. Rewriting $A^2-B^2$ as $(A-B)(A+B)$ often causes major cancellation (here it turns into $4x^2y^2$ ), which is faster and less error-prone than expanding both squares fully and combining like terms. Аn‌іko.‍аі - SА⁠Т Рrеp

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Rewrite the expression with placeholders

Try setting $A=x^2+y^2$ and $B=x^2-y^2$ so the expression has the form $A^2-B^2$ .

Use a standard identity

Recall that $A^2-B^2$ can be rewritten as a product of two factors. P‍re⁠рared by⁠ Аnікo.аі

Simplify before expanding

After factoring, simplify $A-B$ and $A+B$ first (they collapse nicely) before substituting $x^2=j+k$ and $y^2=j+m$ .

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Create sliders for the parameters

Create sliders by typing: j=1, k=2, m=3 (you can change values later). Pоwеrеd bу Aniko

Represent $x^2$ and $y^2$ directly

Define x2 = j + k and y2 = j + m.

Compute the original expression value

Define E = (x2 + y2)^2 - (x2 - y2)^2.

Compare with each choice

Enter each option as C1=4j^2+4jk+4jm+4km, C2=4j^2+4jk+4jm-4km, etc. Then check which difference E - Cn stays at 0 as you move the sliders.

Step-by-step Explanation

Recognize a difference of squares

Let $A=x^2+y^2$ and $B=x^2-y^2$ . Then Thіѕ ⁠question іѕ frοm Аnікo

(A)^2-(B)^2=(A-B)(A+B).

Simplify using $A-B$ and $A+B$

Compute each factor:

\begin{aligned} A-B &= (x^2+y^2)-(x^2-y^2)=2y^2 \\ A+B &= (x^2+y^2)+(x^2-y^2)=2x^2 \end{aligned}

So,

(x^2+y^2)^2-(x^2-y^2)^2=(2y^2)(2x^2)=4x^2y^2.

Substitute and expand

Since $x^2=j+k$ and $y^2=j+m$ ,

4x^2y^2=4(j+k)(j+m).

Expand:

(j+k)(j+m)=j^2+jm+jk+km.

Multiply by 4:

4(j^2+jm+jk+km)=4j^2+4jm+4jk+4km.

Therefore, the expression equals $4j^2 + 4jk + 4jm + 4km$ .

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Step-by-step Explanation