Issolvi 120=3/10^6(x)(24000-x)

$$120 = \frac { 3 } { 10 ^ { 6 } } ( x ) ( 24000 - x )$$

Solvi għal x

$$x=2000\sqrt{26}+12000\approx 22198.039027186$$

$$x=12000-2000\sqrt{26}\approx 1801.960972814$$

Passi għall-Użu tal-Formula Kwadratika

Ikkalkula 10 bil-power ta' 6 u tikseb 1000000.

$$120=\frac{3}{1000000}x\left(24000-x\right)$$

Uża l-propjetà distributtiva biex timmultiplika \frac{3}{1000000}x b'24000-x.

$$120=\frac{3}{1000000}x\times 24000+\frac{3}{1000000}x\left(-1\right)x$$

Immultiplika x u x biex tikseb x^{2}.

$$120=\frac{3}{1000000}x\times 24000+\frac{3}{1000000}x^{2}\left(-1\right)$$

Esprimi \frac{3}{1000000}\times 24000 bħala frazzjoni waħda.

$$120=\frac{3\times 24000}{1000000}x+\frac{3}{1000000}x^{2}\left(-1\right)$$

Immultiplika 3 u 24000 biex tikseb 72000.

$$120=\frac{72000}{1000000}x+\frac{3}{1000000}x^{2}\left(-1\right)$$

Naqqas il-frazzjoni \frac{72000}{1000000} għat-termini l-aktar baxxi billi testratta u tikkanċella barra 8000.

$$120=\frac{9}{125}x+\frac{3}{1000000}x^{2}\left(-1\right)$$

Immultiplika \frac{3}{1000000} u -1 biex tikseb -\frac{3}{1000000}.

$$120=\frac{9}{125}x-\frac{3}{1000000}x^{2}$$

Ibdel in-naħat sabiex it-termini varjabbli kollha jkunu fuq in-naħa tax-xellug.

$$\frac{9}{125}x-\frac{3}{1000000}x^{2}=120$$

Naqqas 120 miż-żewġ naħat.

$$\frac{9}{125}x-\frac{3}{1000000}x^{2}-120=0$$

L-ekwazzjonijiet kollha tal-formola ax^{2}+bx+c=0 jistgħu jiġu solvuti permezz tal-formula kwadratika: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Il-formula kwadratika tagħti żewġ soluzzjonijiet, waħda meta ± hija addizzjoni u waħda meta hija tnaqqis.

$$-\frac{3}{1000000}x^{2}+\frac{9}{125}x-120=0$$

Din l-ekwazzjoni hija fil-forma standard: ax^{2}+bx+c=0. Issostitwixxi -\frac{3}{1000000} għal a, \frac{9}{125} għal b, u -120 għal c fil-formula kwadratika, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\frac{9}{125}±\sqrt{\left(\frac{9}{125}\right)^{2}-4\left(-\frac{3}{1000000}\right)\left(-120\right)}}{2\left(-\frac{3}{1000000}\right)}$$

Ikkwadra \frac{9}{125} billi tikkwadra kemm in-numeratur u d-denominatur tal-frazzjoni.

$$x=\frac{-\frac{9}{125}±\sqrt{\frac{81}{15625}-4\left(-\frac{3}{1000000}\right)\left(-120\right)}}{2\left(-\frac{3}{1000000}\right)}$$

Immultiplika -4 b'-\frac{3}{1000000}.

$$x=\frac{-\frac{9}{125}±\sqrt{\frac{81}{15625}+\frac{3}{250000}\left(-120\right)}}{2\left(-\frac{3}{1000000}\right)}$$

Immultiplika \frac{3}{250000} b'-120.

$$x=\frac{-\frac{9}{125}±\sqrt{\frac{81}{15625}-\frac{9}{6250}}}{2\left(-\frac{3}{1000000}\right)}$$

Żid \frac{81}{15625} ma' -\frac{9}{6250} biex issib id-denominatur komuni u żżid in-numeraturi. Imbagħad naqqas il-frazzjoni għat-termini l-aktar baxxi jekk possibbli.

$$x=\frac{-\frac{9}{125}±\sqrt{\frac{117}{31250}}}{2\left(-\frac{3}{1000000}\right)}$$

Ħu l-għerq kwadrat ta' \frac{117}{31250}.

$$x=\frac{-\frac{9}{125}±\frac{3\sqrt{26}}{250}}{2\left(-\frac{3}{1000000}\right)}$$

Immultiplika 2 b'-\frac{3}{1000000}.

$$x=\frac{-\frac{9}{125}±\frac{3\sqrt{26}}{250}}{-\frac{3}{500000}}$$

Issa solvi l-ekwazzjoni x=\frac{-\frac{9}{125}±\frac{3\sqrt{26}}{250}}{-\frac{3}{500000}} fejn ± hija plus. Żid -\frac{9}{125} ma' \frac{3\sqrt{26}}{250}.

$$x=\frac{\frac{3\sqrt{26}}{250}-\frac{9}{125}}{-\frac{3}{500000}}$$

Iddividi -\frac{9}{125}+\frac{3\sqrt{26}}{250} b'-\frac{3}{500000} billi timmultiplika -\frac{9}{125}+\frac{3\sqrt{26}}{250} bir-reċiproku ta' -\frac{3}{500000}.

$$x=12000-2000\sqrt{26}$$

Issa solvi l-ekwazzjoni x=\frac{-\frac{9}{125}±\frac{3\sqrt{26}}{250}}{-\frac{3}{500000}} fejn ± hija minus. Naqqas \frac{3\sqrt{26}}{250} minn -\frac{9}{125}.

$$x=\frac{-\frac{3\sqrt{26}}{250}-\frac{9}{125}}{-\frac{3}{500000}}$$

Iddividi -\frac{9}{125}-\frac{3\sqrt{26}}{250} b'-\frac{3}{500000} billi timmultiplika -\frac{9}{125}-\frac{3\sqrt{26}}{250} bir-reċiproku ta' -\frac{3}{500000}.

$$x=2000\sqrt{26}+12000$$

L-ekwazzjoni issa solvuta.

$$x=12000-2000\sqrt{26} x=2000\sqrt{26}+12000$$

Passi biex Tikkompleta l-Kwadrat

Ikkalkula 10 bil-power ta' 6 u tikseb 1000000.

$$120=\frac{3}{1000000}x\left(24000-x\right)$$

Uża l-propjetà distributtiva biex timmultiplika \frac{3}{1000000}x b'24000-x.

$$120=\frac{3}{1000000}x\times 24000+\frac{3}{1000000}x\left(-1\right)x$$

Immultiplika x u x biex tikseb x^{2}.

$$120=\frac{3}{1000000}x\times 24000+\frac{3}{1000000}x^{2}\left(-1\right)$$

Esprimi \frac{3}{1000000}\times 24000 bħala frazzjoni waħda.

$$120=\frac{3\times 24000}{1000000}x+\frac{3}{1000000}x^{2}\left(-1\right)$$

Immultiplika 3 u 24000 biex tikseb 72000.

$$120=\frac{72000}{1000000}x+\frac{3}{1000000}x^{2}\left(-1\right)$$

Naqqas il-frazzjoni \frac{72000}{1000000} għat-termini l-aktar baxxi billi testratta u tikkanċella barra 8000.

$$120=\frac{9}{125}x+\frac{3}{1000000}x^{2}\left(-1\right)$$

Immultiplika \frac{3}{1000000} u -1 biex tikseb -\frac{3}{1000000}.

$$120=\frac{9}{125}x-\frac{3}{1000000}x^{2}$$

Ibdel in-naħat sabiex it-termini varjabbli kollha jkunu fuq in-naħa tax-xellug.

$$\frac{9}{125}x-\frac{3}{1000000}x^{2}=120$$

Ekwazzjonijiet kwadratiċi bħal din jistgħu jiġu solvuti billi tikkompleta l-kwadrat. Sabiex tikkompleta l-kwadrat, l-ekwazzjoni l-ewwel trid tkun fil-forma x^{2}+bx=c.

$$-\frac{3}{1000000}x^{2}+\frac{9}{125}x=120$$

Iddividi ż-żewġ naħat tal-ekwazzjoni b'-\frac{3}{1000000}, li hija l-istess bħal multiplikazzjoni taż-żewġ naħat bir-reċiproku tal-frazzjoni.

$$\frac{-\frac{3}{1000000}x^{2}+\frac{9}{125}x}{-\frac{3}{1000000}}=\frac{120}{-\frac{3}{1000000}}$$

Meta tiddividi b'-\frac{3}{1000000} titneħħa l-multiplikazzjoni b'-\frac{3}{1000000}.

$$x^{2}+\frac{\frac{9}{125}}{-\frac{3}{1000000}}x=\frac{120}{-\frac{3}{1000000}}$$

Iddividi \frac{9}{125} b'-\frac{3}{1000000} billi timmultiplika \frac{9}{125} bir-reċiproku ta' -\frac{3}{1000000}.

$$x^{2}-24000x=\frac{120}{-\frac{3}{1000000}}$$

Iddividi 120 b'-\frac{3}{1000000} billi timmultiplika 120 bir-reċiproku ta' -\frac{3}{1000000}.

$$x^{2}-24000x=-40000000$$

Iddividi -24000, il-koeffiċjent tat-terminu x, b'2 biex tikseb -12000. Imbagħad żid il-kwadru ta' -12000 maż-żewġ naħat tal-ekwazzjoni. Dan il-pass jagħmel in-naħa tax-xellug tal-ekwazzjoni kwadru perfett.

$$x^{2}-24000x+\left(-12000\right)^{2}=-40000000+\left(-12000\right)^{2}$$

Ikkwadra -12000.

$$x^{2}-24000x+144000000=-40000000+144000000$$

Żid -40000000 ma' 144000000.

$$x^{2}-24000x+144000000=104000000$$

Fattur x^{2}-24000x+144000000. B'mod ġenerali, meta x^{2}+bx+c huwa kwadru perfett, dejjem jista' jiġu fatturati bħala \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-12000\right)^{2}=104000000$$

Ħu l-għerq kwadrat taż-żewġ naħat tal-ekwazzjoni.

$$\sqrt{\left(x-12000\right)^{2}}=\sqrt{104000000}$$

Issimplifika.

$$x-12000=2000\sqrt{26} x-12000=-2000\sqrt{26}$$

Żid 12000 maż-żewġ naħat tal-ekwazzjoni.

$$x=2000\sqrt{26}+12000 x=12000-2000\sqrt{26}$$