Whakaoti i te 180=3x+x^2

$$180 = 3 x + x ^ { 2 }$$

Whakaoti mō x

$$x=-15$$

$$x=12$$

Ngā Upane Whakamahi Tauwehe

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$3x+x^{2}=180$$

Tangohia te 180 mai i ngā taha e rua.

$$3x+x^{2}-180=0$$

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

$$x^{2}+3x-180=0$$

Hei whakaoti i te whārite, whakatauwehea te x^{2}+3x-180 mā te whakamahi i te tātai x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=3 ab=-180$$

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 -180.

$$-1,180 -2,90 -3,60 -4,45 -5,36 -6,30 -9,20 -10,18 -12,15$$

Tātaihia te tapeke mō ia takirua.

$$-1+180=179 -2+90=88 -3+60=57 -4+45=41 -5+36=31 -6+30=24 -9+20=11 -10+18=8 -12+15=3$$

Ko te otinga te takirua ka hoatu i te tapeke 3.

$$a=-12 b=15$$

Me tuhi anō te kīanga whakatauwehe \left(x+a\right)\left(x+b\right) mā ngā uara i tātaihia.

$$\left(x-12\right)\left(x+15\right)$$

Hei kimi otinga whārite, me whakaoti te x-12=0 me te x+15=0.

$$x=12 x=-15$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$3x+x^{2}=180$$

Tangohia te 180 mai i ngā taha e rua.

$$3x+x^{2}-180=0$$

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

$$x^{2}+3x-180=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx-180. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=3 ab=1\left(-180\right)=-180$$

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 -180.

$$-1,180 -2,90 -3,60 -4,45 -5,36 -6,30 -9,20 -10,18 -12,15$$

Tātaihia te tapeke mō ia takirua.

$$-1+180=179 -2+90=88 -3+60=57 -4+45=41 -5+36=31 -6+30=24 -9+20=11 -10+18=8 -12+15=3$$

Ko te otinga te takirua ka hoatu i te tapeke 3.

$$a=-12 b=15$$

Tuhia anō te x^{2}+3x-180 hei \left(x^{2}-12x\right)+\left(15x-180\right).

$$\left(x^{2}-12x\right)+\left(15x-180\right)$$

Tauwehea te x i te tuatahi me te 15 i te rōpū tuarua.

$$x\left(x-12\right)+15\left(x-12\right)$$

Whakatauwehea atu te kīanga pātahi x-12 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-12\right)\left(x+15\right)$$

Hei kimi otinga whārite, me whakaoti te x-12=0 me te x+15=0.

$$x=12 x=-15$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$3x+x^{2}=180$$

Tangohia te 180 mai i ngā taha e rua.

$$3x+x^{2}-180=0$$

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.

$$x^{2}+3x-180=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 3 mō b, me -180 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-3±\sqrt{3^{2}-4\left(-180\right)}}{2}$$

Pūrua 3.

$$x=\frac{-3±\sqrt{9-4\left(-180\right)}}{2}$$

Whakareatia -4 ki te -180.

$$x=\frac{-3±\sqrt{9+720}}{2}$$

Tāpiri 9 ki te 720.

$$x=\frac{-3±\sqrt{729}}{2}$$

Tuhia te pūtakerua o te 729.

$$x=\frac{-3±27}{2}$$

Nā, me whakaoti te whārite x=\frac{-3±27}{2} ina he tāpiri te ±. Tāpiri -3 ki te 27.

$$x=\frac{24}{2}$$

Whakawehe 24 ki te 2.

$$x=12$$

Nā, me whakaoti te whārite x=\frac{-3±27}{2} ina he tango te ±. Tango 27 mai i -3.

$$x=-\frac{30}{2}$$

Whakawehe -30 ki te 2.

$$x=-15$$

Kua oti te whārite te whakatau.

$$x=12 x=-15$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$3x+x^{2}=180$$

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.

$$x^{2}+3x=180$$

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

$$x^{2}+3x+\left(\frac{3}{2}\right)^{2}=180+\left(\frac{3}{2}\right)^{2}$$

Pūruatia \frac{3}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+3x+\frac{9}{4}=180+\frac{9}{4}$$

Tāpiri 180 ki te \frac{9}{4}.

$$x^{2}+3x+\frac{9}{4}=\frac{729}{4}$$

Tauwehea x^{2}+3x+\frac{9}{4}. 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}.

$$\left(x+\frac{3}{2}\right)^{2}=\frac{729}{4}$$

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

$$\sqrt{\left(x+\frac{3}{2}\right)^{2}}=\sqrt{\frac{729}{4}}$$

Whakarūnātia.

$$x+\frac{3}{2}=\frac{27}{2} x+\frac{3}{2}=-\frac{27}{2}$$

Me tango \frac{3}{2} mai i ngā taha e rua o te whārite.

$$x=12 x=-15$$