SUPER MATH HACKER v2.0

>_ SOLVING POLYNOMIAL EQUATIONS IN CYBERSPACE
>_ USER: STUDENT_LEVEL_10
>_ MISSION: SOLVE 3 EQUATIONS
>_ STATUS: IN_PROGRESS

>_ MYSTERY_QUARTET.EQU

CYBER_DIFFICULTY: 7/10
>_ SOLVE THE ENCRYPTED EQUATION:
(x-5)(x-7)(x+6)(x+4) = 504
1 > // STRATEGIC TERM PAIRING ALGORITHM INITIALIZED
2 > [(x-5)(x+4)] × [(x-7)(x+6)] = 504
3 > // EXPANDING PAIRS...
4 > (x² - x - 20)(x² - x - 42) = 504
5 > // VARIABLE SUBSTITUTION: y = x² - x
6 > (y - 20)(y - 42) = 504
7 > y² - 62y + 840 = 504
8 > y² - 62y + 336 = 0
9 > // SOLVING QUADRATIC EQUATION...
10 > y = [62 ± √(3844 - 1344)]/2 = [62 ± 50]/2
11 > // SOLUTIONS: y = 56 OR y = 6
12 > // REVERTING SUBSTITUTION...
13 > CASE 1: x² - x = 56 → x = 8 || x = -7
14 > CASE 2: x² - x = 6 → x = 3 || x = -2
>_ DECRYPTED SOLUTIONS: x = -7, -2, 3, 8

>_ QUAD_GUARDIANS.EQU

CYBER_DIFFICULTY: 8/10
>_ BREACH THE POLYNOMIAL FIREWALL:
(x-4)(x-7)(x-2)(x+1) = 16
1 > // OPTIMAL PAIRING SEQUENCE DETECTED
2 > [(x-4)(x-2)] × [(x-7)(x+1)] = 16
3 > // EXPANDING MATRIX...
4 > (x² - 6x + 8)(x² - 6x - 7) = 16
5 > // ACTIVATING SUBSTITUTION PROTOCOL: y = x² - 6x
6 > (y + 8)(y - 7) = 16
7 > y² + y - 56 = 16
8 > y² + y - 72 = 0
9 > // EXECUTING QUADRATIC FORMULA...
10 > y = [-1 ± √(1 + 288)]/2 = [-1 ± 17]/2
11 > // SOLUTIONS: y = 8 OR y = -9
12 > // REVERTING TO ORIGINAL VARIABLE...
13 > CASE 1: x² - 6x = 8 → x = 3 ± √17
14 > CASE 2: x² - 6x = -9 → x = 3 (double root)
>_ FIREWALL BREACHED: x = 3, 3 + √17, 3 - √17

>_ MIXED_CHALLENGE.EQU

CYBER_DIFFICULTY: 9/10
>_ CRACK THE ENHANCED ENCRYPTION:
(2x-1)(x+3)(x-2)(2x+3) + 20 = 0
1 > // INITIALIZING ADVANCED HACKING PROTOCOL
2 > [(2x-1)(2x+3)] × [(x+3)(x-2)] + 20 = 0
3 > // EXPANDING TERM CLUSTERS...
4 > (4x² + 4x - 3)(x² + x - 6) + 20 = 0
5 > // SUBSTITUTION ALGORITHM: y = x² + x
6 > (4y - 3)(y - 6) + 20 = 0
7 > 4y² - 27y + 18 + 20 = 0
8 > 4y² - 27y + 38 = 0
9 > // EXECUTING QUADRATIC FORMULA...
10 > y = [27 ± √(729 - 608)]/8 = [27 ± 11]/8
11 > // SOLUTIONS: y = 19/4 OR y = 2
12 > // REVERTING TO BASE VARIABLE...
13 > CASE 1: x² + x = 19/4 → x = (-1 ± 2√5)/2
14 > CASE 2: x² + x = 2 → x = 1 || x = -2
>_ ENCRYPTION CRACKED: x = -2, 1, (-1 + 2√5)/2, (-1 - 2√5)/2
+++ ACCESS GRANTED +++