Calcula la diferencia de los cubos de dos numeros enteros pares consecutivos que es 488 ?

Debemos de saber lo siguiente :

<img src="https://tex.z-dn.net/?f=%28x%29%5E%7B3%7D%20%5CLongleftarrow%20%5Cmathbb%7B%5Ctextbf%7BPRIMER%7D%20%5C%20%5Cmathbb%7BCUBO%7D%20%5C%20PAR%20%5Cmathbb%7B%5Ctextbf%7B%20CONSECUTIVO%7D%7D%7D" />

<img src="https://tex.z-dn.net/?f=%282x%2B2%29%5E%7B3%7D%20%5CLongleftarrow%20%5Cmathbb%7B%5Ctextbf%7BSEGUNDO%7D%20%5C%20%5Cmathbb%7BCUBO%7D%20%5C%20PAR%20%5Cmathbb%7B%5Ctextbf%7B%20CONSECUTIVO%7D%7D%7D" />

Resolvemos.

<img src="https://tex.z-dn.net/?f=%282x%29%5E%7B3%7D-%282x%2B2%29%5E%7B3%7D%3D%20488%20%5C%5C%20%24Calculamos%20el%20cubo%3A%20%5C%5C%20%20%5C%5C%208x%5E%7B3%7D%20-%282x%2B2%29%5E%7B3%7D%3D%20488%20%5C%5C%20%24Factorizamos%20el%20lado%20izquierdo%20y%20expandimos%3A%20%5C%5C%20%20%5C%5C%20%208x%5E%7B3%7D%20-%28%282x%29%5E%7B3%7D%20%2B3%282x%29%5E%7B2%7D%5Ctimes%202%2B3%282x%29%20%5Ctimes2%5E%7B2%7D%20%2B2%5E%7B3%7D%20%29-488%3D%200%20%5C%5C%20%24Movemos%20a%202%20y%20aplicamos%20la%20regla%20del%20producto%20a%202x%3A" />

<img src="https://tex.z-dn.net/?f=8x%5E%7B3%7D%20-%28%282x%29%5E%7B3%7D%20%2B3%282%282%5E%7B2%7Dx%5E%7B2%7D%29%29%2B3%282x%29%20%5Ctimes2%5E%7B2%7D%20%2B2%5E%7B3%7D%20%29-488%3D%200%20%5C%5C%20%24Combinamos%20exponentes%3A%20%5C%5C%20%5C%5C%208x%5E%7B3%7D%20-%28%282x%29%5E%7B3%7D%20%2B3%282%5E%7B2%2B1%7Dx%5E%7B2%7D%29%2B3%282x%29%20%5Ctimes2%5E%7B2%7D%20%2B2%5E%7B3%7D%20%29-488%3D%200%20%5C%5C%24Sumamos%20a%20los%20exponentes%3A%202%2B1%3A%20%5C%5C%20%5C%5C%208x%5E%7B3%7D%20-%28%282x%29%5E%7B3%7D%20%2B3%282%5E%7B3%7Dx%5E%7B2%7D%29%2B3%282x%29%20%5Ctimes2%5E%7B2%7D%20%2B2%5E%7B3%7D%20%29-488%3D0%20%5C%5C%20%24Calculamos%20la%20potencia%3A" />

<img src="https://tex.z-dn.net/?f=8x%5E%7B3%7D%20-%288x%5E%7B3%7D%20%2B3%288x%5E%7B2%7D%29%2B3%282x%29%20%5Ctimes2%5E%7B2%7D%20%2B2%5E%7B3%7D%20%29-488%3D0%20%5C%5C%20%24Lo%20multiplicamos%20a%20%288x%5E%7B2%7D%29%24%20por%203%3A%20%5C%5C%20%5C%5C%208x%5E%7B3%7D%20-%288x%5E%7B3%7D%20%2B24x%5E%7B2%7D%2B3%282x%29%20%5Ctimes2%5E%7B2%7D%20%2B2%5E%7B3%7D%20%29-488%3D0%20%5C%5C%20%24Movemos%20a%202%5E%7B2%7D%3A" />

<img src="https://tex.z-dn.net/?f=8x%5E%7B3%7D%20-%288x%5E%7B3%7D%20%2B24x%5E%7B2%7D%2B3%282%5E%7B2%7D%20%5Ctimes%28%202x%29%29%2B2%5E%7B3%7D%20%29-488%3D0%20%5C%5C%20%24Combinamos%20los%20exponentes%3A%20%5C%5C%20%5C%5C%208x%5E%7B3%7D%20-%288x%5E%7B3%7D%20%2B24x%5E%7B2%7D%2B3%28%202%5E%7B2%2B1%7D%20x%20%29%2B2%5E%7B3%7D%20%29-488%3D%200%5C%5C%20%24Sumamos%20a%20los%20exponentes%3A%202%2B1%3A%20%5C%5C%20%5C%5C%208x%5E%7B3%7D%20-%288x%5E%7B3%7D%20%2B24x%5E%7B2%7D%2B3%28%202%5E%7B3%7D%20x%20%29%2B2%5E%7B3%7D%20%29-488%3D0%20%5C%5C%20%24Calculamos%20la%20potencia%3A%20%5C%5C%20%5C%5C%208x%5E%7B3%7D%20-%288x%5E%7B3%7D%20%2B24x%5E%7B2%7D%2B3%28%208x%20%29%2B2%5E%7B3%7D%20%29-488%3D0%20%5C%5C%20%24Multiplicamos%20a%203%20por%208x%3A" />

<img src="https://tex.z-dn.net/?f=8x%5E%7B3%7D%20-%288x%5E%7B3%7D%20%2B24x%5E%7B2%7D%2B24x%2B2%5E%7B3%7D%20%29-488%3D%20%5C%5C%20%24Calculamos%20la%20potencia%3A%20%5C%5C%20%5C%5C%208x%5E%7B3%7D%20-%288x%5E%7B3%7D%20%2B24x%5E%7B2%7D%2B24x%2B8%20%29-488%3D%200%20%5C%5C%20%24Aplicamos%20la%20Propiedad%20Distributiva%3A%20%5C%5C%20%5C%5C%208x%5E%7B3%7D%20%2B%28-%288x%5E%7B3%7D%29%20-%2824x%5E%7B2%7D%29-%2824x%29%2B8%20%29-488%3D%200" />

<img src="https://tex.z-dn.net/?f=8x%5E%7B3%7D%20%2B%28-%288x%5E%7B3%7D%29%20-%2824x%5E%7B2%7D%29-%2824x%29%2B8%20%29-488%3D%200%20%5C%5C%20%24Simplificamos%20y%20resolvemos%3A%20%5C%5C%20%5C%5C%208x%5E%7B3%7D%20%2B%28-8x%5E%7B3%7D%20-24x%5E%7B2%7D-24x%2B8%20%29-488%3D%200%20%5C%5C%20%5C%5C%208x%5E%7B3%7D%20-8x%5E%7B3%7D%20-24x%5E%7B2%7D-24x%2B8%20-488%3D%200%20%5C%5C%20%5C%5C%20%5Cnot8x%5E%7B3%7D%20-%5Cnot8x%5E%7B3%7D%20-24x%5E%7B2%7D-24x%2B8-488%3D%200%20%5C%5C%20%5C%5C%20-24x%5E%7B2%7D-24x%2B8-488%3D%200%20%5C%5C%20%24Dividimos%20entre%208%3A" />

<img src="https://tex.z-dn.net/?f=3x%5E2%20%2B%203x%20%2B%201%20-61%3D%200%20%5C%5C%20%20%5C%5C%203x%5E2%20%2B%203x-60%3D%200%20%5C%5C%20%24Factorizamos%20usando%20a%20%223%22%3A" />

<img src="https://tex.z-dn.net/?f=3%28%20x%5E%7B2%7D%20%2Bx-20%29%3D%200%20%5C%5C%20%24Dividimos%20a%203%20entre%20la%20parte%20derecha%20de%200%3A%20%5C%5C%20%20%5C%5C%20%28x%5E%7B2%7D%20%2Bx-20%29%3D%20%20%5Cdfrac%7B0%7D%7B3%7D%20%20%5C%5C%20%20%5C%5C%20%28x%5E%7B2%7D%20%2Bx-20%29%3D%200%20%5C%5C%20%24Buscamos%20un%20numero%20que%20sumado%20de%20x%2C%20es%20decir%20%221%22%20y%20que%20su%20producto%20%20%5C%5C%20sea%20%20de%20-20%3A%20%5C%5C%20%20%5C%5C%20Esos%20n%5C%27umeros%20son%3A%20%5Ctextbf%7B-4%3B%205.%7D%20%5C%5C%20%20%5C%5C%20%28x-4%29%28x%2B5%29%3D%200%20%5C%5C%20Despejamos%20la%20parte%20izquierda%20de%20la%20ecuaci%5C%27on%20y%20pasamos%20a%20dividir%20la%20%20%5C%5C%20parte%20derecha%3A%20%5C%5C%20%20%5C%5C%20%28x-4%29%3D%20%20%5Cdfrac%7B0%7D%7B%28x%2B5%29%7D%20%20%5C%5C%20%20%5C%5C%20x-4%3D%200%20%5C%5C%20%20%5C%5C%20%5Cboxed%7B%5Cboxed%7B%5Ctextbf%7Bx%3D%204%7D%7D%7D%20%5Ccheckmark" />

[img = 10]

Los valores de "x" son : 4 ; - 5.

Por lo tanto, los númerosson 8 y 10, o - 8 y - 10.

Ya que encontramos sus valores así :

[img = 11]

Los númerospares consecutivos son : 8 ; 10, o - 8 ; - 10.

Comprobaremos sustituyendo los datos :

[img = 12]

[img = 13]

Saludos y Suerte!