Μαθηματικά Γ' Γυμ. : Θεωρία: Ταυτότητες: Αξιοσημείωτες ταυτότητες

Θεωρία: Ταυτότητες

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Αξιοσημείωτες ταυτότητες

Ταυτότητα λέγεται κάθε ισότητα που περιέχει μεταβλητές και αληθεύει για όλες τις τιμές των μεταβλητών της.

Αξιοσημείωτες ταυτότητες

Το δεύτερα μέλη των ταυτοτήτων που ακολουθούν ονομάζονται αναπτύγματα.

Τετράγωνο αθροίσματος

$open parentheses alpha blank plus blank beta close parentheses squared equals blank alpha squared plus blank 2 alpha beta blank plus blank beta squared$

$open parentheses y blank plus blank 4 close parentheses squared equals blank y squared plus blank 2 times y times 4 blank plus blank 4 squared equals blank y squared plus blank 8 y blank plus blank 16$

$open parentheses square root of 3 space plus 1 close parentheses squared equals open parentheses square root of 3 close parentheses squared plus blank 2 times square root of 3 space times 1 blank plus blank straight 1 squared equals blank 3 plus blank 2 square root of 3 blank plus blank 1 equals 4 plus 2 square root of 3 blank$

Τετράγωνο διαφοράς

$open parentheses alpha blank minus blank beta close parentheses squared equals blank alpha squared minus blank 2 alpha beta blank plus blank beta squared$

$open parentheses omega space minus 2 over omega close parentheses squared equals omega squared minus 2 times omega space times 2 over omega plus open parentheses 2 over omega close parentheses squared equals omega squared minus 4 plus 4 over omega squared$

$open parentheses 1 minus square root of 7 close parentheses squared equals 1 minus 2 times 1 times square root of 7 space plus open parentheses square root of 7 close parentheses squared equals blank 1 minus blank 2 square root of 7 blank plus straight 7 equals 8 minus 2 square root of 7 blank$

Κύβος αθροίσματος

$open parentheses alpha blank plus blank beta close parentheses cubed equals blank alpha cubed plus blank 3 alpha squared beta space plus 3 alpha beta squared plus beta cubed$

$open parentheses straight 2 straight x plus straight 1 close parentheses cubed equals blank open parentheses 2 x close parentheses cubed plus blank 3 times open parentheses 2 x close parentheses squared times 1 plus 3 times open parentheses 2 x close parentheses times 1 squared plus 1 cubed equals 8 x cubed plus 12 x squared plus 6 x plus 1$

$open parentheses square root of straight 2 blank plus straight 1 close parentheses cubed equals blank open parentheses square root of straight 2 close parentheses cubed plus blank 3 times open parentheses square root of straight 2 close parentheses squared times 1 plus 3 times open parentheses square root of straight 2 close parentheses times 1 squared plus 1 cubed equals open parentheses square root of straight 2 close parentheses squared times square root of straight 2 space plus 3 times 2 times 1 plus 3 square root of 2 space plus 1 equals 2 square root of straight 2 space plus 6 plus 3 square root of 2 space plus 1 equals 5 square root of straight 2 space plus 7$

Κύβος διαφοράς

$open parentheses alpha blank minus blank beta close parentheses cubed equals blank alpha cubed minus blank 3 alpha squared beta space plus 3 alpha beta squared minus beta cubed$

$open parentheses omega squared minus blank 2 omega close parentheses cubed equals open parentheses omega squared close parentheses cubed minus blank 3 times open parentheses omega squared close parentheses squared times open parentheses 2 omega close parentheses plus 3 times open parentheses omega squared close parentheses times open parentheses 2 omega close parentheses squared minus open parentheses 2 omega close parentheses cubed equals omega to the power of 6 minus 3 times open parentheses omega to the power of 4 close parentheses times open parentheses 2 omega close parentheses plus 3 times open parentheses omega squared close parentheses times open parentheses 4 omega squared close parentheses minus 8 omega cubed equals omega to the power of 6 minus 6 omega to the power of 5 plus 12 omega to the power of 4 minus 8 omega cubed$

$open parentheses square root of straight 2 straight space straight minus straight space square root of straight 3 close parentheses cubed equals blank open parentheses square root of straight 2 close parentheses cubed straight minus straight space 3 times open parentheses square root of straight 2 close parentheses squared times square root of straight 3 space plus 3 times open parentheses square root of straight 2 close parentheses times open parentheses square root of straight 3 close parentheses squared minus open parentheses square root of straight 3 close parentheses cubed equals open parentheses square root of straight 2 close parentheses squared times square root of straight 2 space minus 3 times 2 times square root of straight 3 space plus 3 times square root of 2 space times 3 minus open parentheses square root of straight 3 close parentheses squared times square root of straight 3 space equals 2 square root of straight 2 space minus 6 square root of straight 3 space plus 9 square root of 2 space minus 3 square root of straight 3 space equals 11 square root of 2 space minus 9 square root of straight 3$

Γινόμενο αθροίσματος επί διαφορά

$open parentheses alpha straight space straight plus straight space beta close parentheses open parentheses alpha blank minus blank beta close parentheses straight space straight equals straight space alpha squared minus blank beta squared$

$open parentheses alpha cubed straight space straight plus straight space beta cubed close parentheses open parentheses alpha cubed blank minus blank beta cubed close parentheses straight space straight equals straight space open parentheses alpha cubed close parentheses squared minus blank open parentheses beta cubed close parentheses squared equals alpha to the power of 6 blank minus blank beta to the power of 6$

$99 times 101 equals open parentheses 100 minus 1 close parentheses open parentheses 100 plus 1 close parentheses equals 100 squared minus 1 squared equals 10000 minus 1 equals 9999$