Riješi x^2/(4-x)(1-x)=13*10^9

$$\frac { x ^ { 2 } } { ( 4 - x ) ( 1 - x ) } = 13 \times 10 ^ { 9 }$$

Izračunaj x

$$x = \frac{20000 \sqrt{950625000130} + 32500000000}{12999999999} \approx 4$$

$$x=\frac{32500000000-20000\sqrt{950625000130}}{12999999999}\approx 1$$

Koraci korištenja kvadratne formule

Varijabla x ne može biti jednaka vrijednostima 1,4 jer nije definirano dijeljenje nulom. Pomnožite obje strane jednadžbe s \left(x-4\right)\left(x-1\right).

$$x^{2}=13\times 10^{9}\left(x-4\right)\left(x-1\right)$$

Izračunajte koliko je 9 na 10 da biste dobili 1000000000.

$$x^{2}=13\times 1000000000\left(x-4\right)\left(x-1\right)$$

Pomnožite 13 i 1000000000 da biste dobili 13000000000.

$$x^{2}=13000000000\left(x-4\right)\left(x-1\right)$$

Koristite svojstvo distributivnosti da biste pomnožili 13000000000 s x-4.

$$x^{2}=\left(13000000000x-52000000000\right)\left(x-1\right)$$

Koristite svojstvo distributivnosti da biste pomnožili 13000000000x-52000000000 s x-1 i kombinirali slične izraze.

$$x^{2}=13000000000x^{2}-65000000000x+52000000000$$

Oduzmite 13000000000x^{2} od obiju strana.

$$x^{2}-13000000000x^{2}=-65000000000x+52000000000$$

Kombinirajte x^{2} i -13000000000x^{2} da biste dobili -12999999999x^{2}.

$$-12999999999x^{2}=-65000000000x+52000000000$$

Dodajte 65000000000x na obje strane.

$$-12999999999x^{2}[phone]x=52000000000$$

Oduzmite 52000000000 od obiju strana.

$$-12999999999x^{2}[phone]x-52000000000=0$$

Ova je jednadžba u standardnom obliku: ax^{2}+bx+c=0. U kvadratnoj jednadžbi \frac{-b±\sqrt{b^{2}-4ac}}{2a} zamijenite -12999999999 s a, 65000000000 s b i -52000000000 s c.

$$x=\frac{-65000000000±\sqrt{65000000000^{2}-4\left(-12999999999\right)\left(-52000000000\right)}}{2\left(-12999999999\right)}$$

Kvadrirajte 65000000000.

$$x=\frac{-65000000000±\sqrt{4225000000000000000000-4\left(-12999999999\right)\left(-52000000000\right)}}{2\left(-12999999999\right)}$$

Pomnožite -4 i -12999999999.

$$x=\frac{-65000000000±\sqrt{4225000000000000000000+51999999996\left(-52000000000\right)}}{2\left(-12999999999\right)}$$

Pomnožite 51999999996 i -52000000000.

$$x=\frac{-65000000000±\sqrt{4225000000000000000000-2703999999792000000000}}{2\left(-12999999999\right)}$$

Dodaj 4225000000000000000000 broju -2703999999792000000000.

$$x=\frac{-65000000000±\sqrt{1521000000208000000000}}{2\left(-12999999999\right)}$$

Izračunajte kvadratni korijen od 1521000000208000000000.

$$x=\frac{-65000000000±40000\sqrt{950625000130}}{2\left(-12999999999\right)}$$

Pomnožite 2 i -12999999999.

$$x=\frac{-65000000000±40000\sqrt{950625000130}}{-25999999998}$$

Sada riješite jednadžbu x=\frac{-65000000000±40000\sqrt{950625000130}}{-25999999998} kad je ± plus. Dodaj -65000000000 broju 40000\sqrt{950625000130}.

$$x=\frac{40000\sqrt{950625000130}-65000000000}{-25999999998}$$

Podijelite -65000000000+40000\sqrt{950625000130} s -25999999998.

$$x=\frac{32500000000-20000\sqrt{950625000130}}{12999999999}$$

Sada riješite jednadžbu x=\frac{-65000000000±40000\sqrt{950625000130}}{-25999999998} kad je ± minus. Oduzmite 40000\sqrt{950625000130} od -65000000000.

$$x=\frac{-40000\sqrt{950625000130}-65000000000}{-25999999998}$$

Podijelite -65000000000-40000\sqrt{950625000130} s -25999999998.

$$x=\frac{20000\sqrt{950625000130}[phone]}{12999999999}$$

Jednadžba je sada riješena.

$$x=\frac{32500000000-20000\sqrt{950625000130}}{12999999999} x=\frac{20000\sqrt{950625000130}[phone]}{12999999999}$$

Koraci za računanje kvadrata

Varijabla x ne može biti jednaka vrijednostima 1,4 jer nije definirano dijeljenje nulom. Pomnožite obje strane jednadžbe s \left(x-4\right)\left(x-1\right).

$$x^{2}=13\times 10^{9}\left(x-4\right)\left(x-1\right)$$

Izračunajte koliko je 9 na 10 da biste dobili 1000000000.

$$x^{2}=13\times 1000000000\left(x-4\right)\left(x-1\right)$$

Pomnožite 13 i 1000000000 da biste dobili 13000000000.

$$x^{2}=13000000000\left(x-4\right)\left(x-1\right)$$

Koristite svojstvo distributivnosti da biste pomnožili 13000000000 s x-4.

$$x^{2}=\left(13000000000x-52000000000\right)\left(x-1\right)$$

Koristite svojstvo distributivnosti da biste pomnožili 13000000000x-52000000000 s x-1 i kombinirali slične izraze.

$$x^{2}=13000000000x^{2}-65000000000x+52000000000$$

Oduzmite 13000000000x^{2} od obiju strana.

$$x^{2}-13000000000x^{2}=-65000000000x+52000000000$$

Kombinirajte x^{2} i -13000000000x^{2} da biste dobili -12999999999x^{2}.

$$-12999999999x^{2}=-65000000000x+52000000000$$

Dodajte 65000000000x na obje strane.

$$-12999999999x^{2}[phone]x=52000000000$$

Podijelite obje strane sa -12999999999.

$$\frac{-12999999999x^{2}[phone]x}{-12999999999}=\frac{52000000000}{-12999999999}$$

Dijeljenjem s -12999999999 poništava se množenje s -12999999999.

$$x^{2}+\frac{65000000000}{-12999999999}x=\frac{52000000000}{-12999999999}$$

Podijelite 65000000000 s -12999999999.

$$x^{2}-\frac{65000000000}{12999999999}x=\frac{52000000000}{-12999999999}$$

Podijelite 52000000000 s -12999999999.

$$x^{2}-\frac{65000000000}{12999999999}x=-\frac{52000000000}{12999999999}$$

Podijelite -\frac{65000000000}{12999999999}, koeficijent izraza x, s 2 da biste dobili -\frac{32500000000}{12999999999}. Zatim objema stranama jednadžbe pridodajte -\frac{32500000000}{12999999999} na kvadrat. Tim korakom lijeva strana jednadžbe postaje potpuna kvadratna jednadžba.

$$x^{2}-\frac{65000000000}{12999999999}x+\left(-\frac{32500000000}{12999999999}\right)^{2}=-\frac{52000000000}{12999999999}+\left(-\frac{32500000000}{12999999999}\right)^{2}$$

Kvadrirajte -\frac{32500000000}{12999999999} tako da kvadrirate brojnik i nazivnik razlomka.

$$x^{2}-\frac{65000000000}{12999999999}x+\frac{1056250000000000000000}{168999999974000000001}=-\frac{52000000000}{12999999999}+\frac{1056250000000000000000}{168999999974000000001}$$

Dodajte -\frac{52000000000}{12999999999} broju \frac{1056250000000000000000}{168999999974000000001} pronalaženjem zajedničkog nazivnika i zbrajanjem brojnika. Zatim pokratite razlomak ako je to moguće.

$$x^{2}-\frac{65000000000}{12999999999}x+\frac{1056250000000000000000}{168999999974000000001}=\frac{380250000052000000000}{168999999974000000001}$$

Faktor x^{2}-\frac{65000000000}{12999999999}x+\frac{1056250000000000000000}{168999999974000000001}. Općenito, kada je x^{2}+bx+c savršen kvadrat, uvijek se može uzeti u obzir kao \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{32500000000}{12999999999}\right)^{2}=\frac{380250000052000000000}{168999999974000000001}$$

Izračunajte kvadratni korijen obiju strana jednadžbe.

$$\sqrt{\left(x-\frac{32500000000}{12999999999}\right)^{2}}=\sqrt{\frac{380250000052000000000}{168999999974000000001}}$$

Pojednostavnite.

$$x-\frac{32500000000}{12999999999}=\frac{20000\sqrt{950625000130}}{12999999999} x-\frac{32500000000}{12999999999}=-\frac{20000\sqrt{950625000130}}{12999999999}$$

Dodajte \frac{32500000000}{12999999999} objema stranama jednadžbe.

$$x=\frac{20000\sqrt{950625000130}[phone]}{12999999999} x=\frac{32500000000-20000\sqrt{950625000130}}{12999999999}$$