Whakaoti i te x+3x^2=520

$$x+3 { x }^{ 2 } =520$$

Whakaoti mō x

$$x = -\frac{40}{3} = -13\frac{1}{3} \approx -13.333333333$$

$$x=13$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tangohia te 520 mai i ngā taha e rua.

$$x+3x^{2}-520=0$$

Hurinahatia te pūrau ki te āhua tānga ngahuru. Whakaraupapahia ngā kīanga tau mai i te pū teitei rawa ki te mea iti rawa.

$$3x^{2}+x-520=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 3x^{2}+ax+bx-520. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=1 ab=3\left(-520\right)=-1560$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōrunga te a+b, he nui ake te uara pū o te tau tōrunga i tō te tōraro. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -1560.

$$-1,1560 -2,780 -3,520 -4,390 -5,312 -6,260 -8,195 -10,156 -12,130 -13,120 -15,104 -20,78 -24,65 -26,60 -30,52 -39,40$$

Tātaihia te tapeke mō ia takirua.

$$-1+1560=1559 -2+780=778 -3+520=517 -4+390=386 -5+312=307 -6+260=254 -8+195=187 -10+156=146 -12+130=118 -13+120=107 -15+104=89 -20+78=58 -24+65=41 -26+60=34 -30+52=22 -39+40=1$$

Ko te otinga te takirua ka hoatu i te tapeke 1.

$$a=-39 b=40$$

Tuhia anō te 3x^{2}+x-520 hei \left(3x^{2}-39x\right)+\left(40x-520\right).

$$\left(3x^{2}-39x\right)+\left(40x-520\right)$$

Tauwehea te 3x i te tuatahi me te 40 i te rōpū tuarua.

$$3x\left(x-13\right)+40\left(x-13\right)$$

Whakatauwehea atu te kīanga pātahi x-13 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-13\right)\left(3x+40\right)$$

Hei kimi otinga whārite, me whakaoti te x-13=0 me te 3x+40=0.

$$x=13 x=-\frac{40}{3}$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Ko ngā whārite katoa o te āhua ax^{2}+bx+c=0 ka taea te whakaoti mā te whakamahi i te tikanga tātai pūrua: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. E rua ngā otinga ka puta i te tikanga tātai pūrua, ko tētahi ina he tāpiri a ±, ā, ko tētahi ina he tango.

$$3x^{2}+x=520$$

Me tango 520 mai i ngā taha e rua o te whārite.

$$3x^{2}+x-520=520-520$$

Mā te tango i te 520 i a ia ake anō ka toe ko te 0.

$$3x^{2}+x-520=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 3 mō a, 1 mō b, me -520 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1±\sqrt{1^{2}-4\times 3\left(-520\right)}}{2\times 3}$$

Pūrua 1.

$$x=\frac{-1±\sqrt{1-4\times 3\left(-520\right)}}{2\times 3}$$

Whakareatia -4 ki te 3.

$$x=\frac{-1±\sqrt{1-12\left(-520\right)}}{2\times 3}$$

Whakareatia -12 ki te -520.

$$x=\frac{-1±\sqrt{1+6240}}{2\times 3}$$

Tāpiri 1 ki te 6240.

$$x=\frac{-1±\sqrt{6241}}{2\times 3}$$

Tuhia te pūtakerua o te 6241.

$$x=\frac{-1±79}{2\times 3}$$

Whakareatia 2 ki te 3.

$$x=\frac{-1±79}{6}$$

Nā, me whakaoti te whārite x=\frac{-1±79}{6} ina he tāpiri te ±. Tāpiri -1 ki te 79.

$$x=\frac{78}{6}$$

Whakawehe 78 ki te 6.

$$x=13$$

Nā, me whakaoti te whārite x=\frac{-1±79}{6} ina he tango te ±. Tango 79 mai i -1.

$$x=-\frac{80}{6}$$

Whakahekea te hautanga \frac{-80}{6} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x=-\frac{40}{3}$$

Kua oti te whārite te whakatau.

$$x=13 x=-\frac{40}{3}$$

Ngā Upane mō te Whakaoti i te Pūrua

Ko ngā whārite pūrua pēnei i tēnei nā ka taea te whakaoti mā te whakaoti i te pūrua. Hei whakaoti i te pūrua, ko te whārite me mātua tuhi ki te āhua x^{2}+bx=c.

$$3x^{2}+x=520$$

Whakawehea ngā taha e rua ki te 3.

$$\frac{3x^{2}+x}{3}=\frac{520}{3}$$

Mā te whakawehe ki te 3 ka wetekia te whakareanga ki te 3.

$$x^{2}+\frac{1}{3}x=\frac{520}{3}$$

Whakawehea te \frac{1}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{1}{6}. Nā, tāpiria te pūrua o te \frac{1}{6} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{1}{3}x+\left(\frac{1}{6}\right)^{2}=\frac{520}{3}+\left(\frac{1}{6}\right)^{2}$$

Pūruatia \frac{1}{6} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{1}{3}x+\frac{1}{36}=\frac{520}{3}+\frac{1}{36}$$

Tāpiri \frac{520}{3} ki te \frac{1}{36} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x^{2}+\frac{1}{3}x+\frac{1}{36}=\frac{6241}{36}$$

Tauwehea x^{2}+\frac{1}{3}x+\frac{1}{36}. Ko te tikanga pūnoa, ina ko x^{2}+bx+c he pūrua tika pūrua tika pū, ka taea taua mea te tauwehea i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{1}{6}\right)^{2}=\frac{6241}{36}$$

Tuhia te pūtakerua o ngā taha e rua o te whārite.

$$\sqrt{\left(x+\frac{1}{6}\right)^{2}}=\sqrt{\frac{6241}{36}}$$

Whakarūnātia.

$$x+\frac{1}{6}=\frac{79}{6} x+\frac{1}{6}=-\frac{79}{6}$$

Me tango \frac{1}{6} mai i ngā taha e rua o te whārite.

$$x=13 x=-\frac{40}{3}$$