Rešite x^2/(4-x)(1-x)=13*10^9

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

Rešitev za x

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

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

Koraki z uporabo formule za reševanje kvadratne enačbe

Spremenljivka x ne more biti enaka nobeni od vrednosti 1,4, ker deljenje z vrednostjo nič ni določeno. Pomnožite obe strani enačbe s/z \left(x-4\right)\left(x-1\right).

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

Izračunajte potenco 10 števila 9, da dobite 1000000000.

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

Pomnožite 13 in 1000000000, da dobite 13000000000.

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

Uporabite distributivnost, da pomnožite 13000000000 s/z x-4.

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

Uporabite lastnost distributivnosti za množenje 13000000000x-52000000000 krat x-1 in kombiniranje pogojev podobnosti.

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

Odštejte 13000000000x^{2} na obeh straneh.

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

Združite x^{2} in -13000000000x^{2}, da dobite -12999999999x^{2}.

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

Dodajte 65000000000x na obe strani.

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

Odštejte 52000000000 na obeh straneh.

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

Ta enačba je v standardni obliki: ax^{2}+bx+c=0. Vstavite -12999999999 za a, 65000000000 za b in -52000000000 za c v formulo za reševanje kvadratnih enačb \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Kvadrat števila 65000000000.

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

Pomnožite -4 s/z -12999999999.

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

Pomnožite 51999999996 s/z -52000000000.

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

Seštejte 4225000000000000000000 in -2703999999792000000000.

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

Uporabite kvadratni koren števila 1521000000208000000000.

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

Pomnožite 2 s/z -12999999999.

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

Zdaj rešite enačbo x=\frac{-65000000000±40000\sqrt{950625000130}}{-25999999998}, ko je ± plus. Seštejte -65000000000 in 40000\sqrt{950625000130}.

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

Delite -65000000000+40000\sqrt{950625000130} s/z -25999999998.

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

Zdaj rešite enačbo x=\frac{-65000000000±40000\sqrt{950625000130}}{-25999999998}, ko je ± minus. Odštejte 40000\sqrt{950625000130} od -65000000000.

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

Delite -65000000000-40000\sqrt{950625000130} s/z -25999999998.

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

Enačba je zdaj rešena.

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

Koraki za dopolnjevanje do popolnega kvadrata

Spremenljivka x ne more biti enaka nobeni od vrednosti 1,4, ker deljenje z vrednostjo nič ni določeno. Pomnožite obe strani enačbe s/z \left(x-4\right)\left(x-1\right).

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

Izračunajte potenco 10 števila 9, da dobite 1000000000.

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

Pomnožite 13 in 1000000000, da dobite 13000000000.

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

Uporabite distributivnost, da pomnožite 13000000000 s/z x-4.

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

Uporabite lastnost distributivnosti za množenje 13000000000x-52000000000 krat x-1 in kombiniranje pogojev podobnosti.

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

Odštejte 13000000000x^{2} na obeh straneh.

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

Združite x^{2} in -13000000000x^{2}, da dobite -12999999999x^{2}.

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

Dodajte 65000000000x na obe strani.

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

Delite obe strani z vrednostjo -12999999999.

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

Z deljenjem s/z -12999999999 razveljavite množenje s/z -12999999999.

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

Delite 65000000000 s/z -12999999999.

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

Delite 52000000000 s/z -12999999999.

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

Delite -\frac{65000000000}{12999999999}, ki je koeficient člena x, z 2, da dobite -\frac{32500000000}{12999999999}. Nato dodajte kvadrat števila -\frac{32500000000}{12999999999} na obe strani enačbe. S tem korakom boste levo stran enačbe pretvorili v popolni kvadrat.

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

Kvadrirajte ulomek -\frac{32500000000}{12999999999} tako, da kvadrirate števec in imenovalec ulomka.

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

Seštejte -\frac{52000000000}{12999999999} in \frac{1056250000000000000000}{168999999974000000001} tako, da poiščete skupni imenovalec in seštejete števce. Nato okrajšajte ulomek do najnižjih možnih členov.

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

Faktorizirajte x^{2}-\frac{65000000000}{12999999999}x+\frac{1056250000000000000000}{168999999974000000001}. Če je x^{2}+bx+c kvadrat, ga lahko vedno faktorizirate kot \left(x+\frac{b}{2}\right)^{2}.

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

Uporabite kvadratni koren obeh strani enačbe.

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

Poenostavite.

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

Prištejte \frac{32500000000}{12999999999} na obe strani enačbe.

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