Riješi x^2/(4-x)(1-x)=13*10^9

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

Riješite za x

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

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

Koraci pomoću kvadratne formule

Promjenjiva x ne može biti jednaka nijednoj od vrijednosti 1,4 zato što dijeljenje nulom nije definirano. Pomnožite obje strane jednačine sa \left(x-4\right)\left(x-1\right).

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

Izračunajte 10 stepen od 9 i dobijte 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 distributivno svojstvo da biste pomnožili 13000000000 sa 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 pojmove.

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

Oduzmite 13000000000x^{2} s obje strane.

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

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 s obje strane.

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

Ova jednačina je u standardnom obliku: ax^{2}+bx+c=0. Zamijenite -12999999999 i a, 65000000000 i b, kao i -52000000000 i c u kvadratnoj formuli, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Izračunajte kvadrat od 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.

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

Saberite 4225000000000000000000 i -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 jednačinu x=\frac{-65000000000±40000\sqrt{950625000130}}{-25999999998} kada je ± plus. Saberite -65000000000 i 40000\sqrt{950625000130}.

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

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

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

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

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

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

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

Jednačina je riješena.

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

Koraci za dovršavanje kvadrata

Promjenjiva x ne može biti jednaka nijednoj od vrijednosti 1,4 zato što dijeljenje nulom nije definirano. Pomnožite obje strane jednačine sa \left(x-4\right)\left(x-1\right).

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

Izračunajte 10 stepen od 9 i dobijte 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 distributivno svojstvo da biste pomnožili 13000000000 sa 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 pojmove.

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

Oduzmite 13000000000x^{2} s obje strane.

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

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 s -12999999999.

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

Dijelјenje sa -12999999999 poništava množenje sa -12999999999.

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

Podijelite 65000000000 sa -12999999999.

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

Podijelite 52000000000 sa -12999999999.

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

Podijelite -\frac{65000000000}{12999999999}, koeficijent izraza x, sa 2 da biste dobili -\frac{32500000000}{12999999999}. Zatim dodajte kvadrat od -\frac{32500000000}{12999999999} na obje strane jednačine. Ovaj korak čini lijevu stranu jednačine savršenim kvadratom.

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

Izračunajte kvadrat od -\frac{32500000000}{12999999999} tako što ćete izračunati kvadrat od brojioca i imenioca razlomka.

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

Saberite -\frac{52000000000}{12999999999} i \frac{1056250000000000000000}{168999999974000000001} tako što ćete pronaći zajednički imenilac i sabrati brojioce. Zatim svedite razlomak na najniže termine ukoliko je moguće.

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

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

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

Izračunajte kvadratni korijen od obje strane jednačine.

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

Pojednostavite.

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

Dodajte \frac{32500000000}{12999999999} na obje strane jednačine.

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