Skillnad mellan versioner av "1.4 Lösning 11"
Taifun (Diskussion | bidrag) m |
Taifun (Diskussion | bidrag) m |
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(4 mellanliggande versioner av samma användare visas inte) | |||
Rad 3: | Rad 3: | ||
:<math> {2\,x^2 - x^3 \over 2\,x^2 - 8} \; = \; {x^2\,(2 - x) \over 2\,(x^2 - 4)} \; = \; {x^2\,(2 - x) \over 2\,(x-2)\cdot(x+2)} \; = \; </math> | :<math> {2\,x^2 - x^3 \over 2\,x^2 - 8} \; = \; {x^2\,(2 - x) \over 2\,(x^2 - 4)} \; = \; {x^2\,(2 - x) \over 2\,(x-2)\cdot(x+2)} \; = \; </math> | ||
+ | :<math> = \; {-\,x^2\,(x - 2) \over 2\,(x-2)\cdot(x+2)} \; = \; {-\,x^2 \over 2\,(x+2)} </math> | ||
− | + | Detta sätts in i hela uttrycket: | |
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:<math> {2\,x^2 - x^3 \over 2\,x^2 - 8} - {x \over x+2} + {x+2 \over 2} \; = \; {-\,x^2 \over 2\,(x+2)} - {x \over x+2} + {x+2 \over 2} \; = \; </math> | :<math> {2\,x^2 - x^3 \over 2\,x^2 - 8} - {x \over x+2} + {x+2 \over 2} \; = \; {-\,x^2 \over 2\,(x+2)} - {x \over x+2} + {x+2 \over 2} \; = \; </math> | ||
+ | :<math> = \; {-\,x^2 \over 2\,(x+2)} - {2\cdot x \over 2\cdot(x+2)} + {(x+2)\cdot(x+2) \over 2\cdot(x+2)} \; = \; </math> | ||
− | + | :<math> = \; {-\,x^2 - 2\,x + (x+2)^2 \over 2\,(x+2)} \; = \; {-\,x^2 - 2\,x + x^2 + 4\,x +4 \over 2\,(x+2)} \; = \; </math> | |
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− | <math> = \; {-\,x^2 - 2\,x + (x+2)^2 \over 2\,(x+2)} \; = \; {-\,x^2 - 2\,x + x^2 + 4\,x +4 \over 2\,(x+2)} \; = \; </math> | + | |
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− | <math> = \; {-2\,x + 4\,x + 4 \over 2\,(x+2)} \; = {2\,x + 4 \over 2\,(x+2)} \; = \; {2\,(x+2) \over 2\,(x+2)} \; = \; 1 </math> | + | :<math> = \; {-2\,x + 4\,x + 4 \over 2\,(x+2)} \; = {2\,x + 4 \over 2\,(x+2)} \; = \; {2\,(x+2) \over 2\,(x+2)} \; = \; 1 </math> |
Nuvarande version från 3 augusti 2014 kl. 23.53
Vi förenklar först uttryckets första term:
\[ {2\,x^2 - x^3 \over 2\,x^2 - 8} \; = \; {x^2\,(2 - x) \over 2\,(x^2 - 4)} \; = \; {x^2\,(2 - x) \over 2\,(x-2)\cdot(x+2)} \; = \; \]
\[ = \; {-\,x^2\,(x - 2) \over 2\,(x-2)\cdot(x+2)} \; = \; {-\,x^2 \over 2\,(x+2)} \]
Detta sätts in i hela uttrycket:
\[ {2\,x^2 - x^3 \over 2\,x^2 - 8} - {x \over x+2} + {x+2 \over 2} \; = \; {-\,x^2 \over 2\,(x+2)} - {x \over x+2} + {x+2 \over 2} \; = \; \]
\[ = \; {-\,x^2 \over 2\,(x+2)} - {2\cdot x \over 2\cdot(x+2)} + {(x+2)\cdot(x+2) \over 2\cdot(x+2)} \; = \; \]
\[ = \; {-\,x^2 - 2\,x + (x+2)^2 \over 2\,(x+2)} \; = \; {-\,x^2 - 2\,x + x^2 + 4\,x +4 \over 2\,(x+2)} \; = \; \]
\[ = \; {-2\,x + 4\,x + 4 \over 2\,(x+2)} \; = {2\,x + 4 \over 2\,(x+2)} \; = \; {2\,(x+2) \over 2\,(x+2)} \; = \; 1 \]