Whakaoti i te 3x^2+9x+6=90

$$3 x ^ { 2 } + 9 x + 6 = 90$$

Whakaoti mō x

$$x=-7$$

$$x=4$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tangohia te 90 mai i ngā taha e rua.

$$3x^{2}+9x+6-90=0$$

Tangohia te 90 i te 6, ka -84.

$$3x^{2}+9x-84=0$$

Whakawehea ngā taha e rua ki te 3.

$$x^{2}+3x-28=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx-28. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=3 ab=1\left(-28\right)=-28$$

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 -28.

$$-1,28 -2,14 -4,7$$

Tātaihia te tapeke mō ia takirua.

$$-1+28=27 -2+14=12 -4+7=3$$

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

$$a=-4 b=7$$

Tuhia anō te x^{2}+3x-28 hei \left(x^{2}-4x\right)+\left(7x-28\right).

$$\left(x^{2}-4x\right)+\left(7x-28\right)$$

Tauwehea te x i te tuatahi me te 7 i te rōpū tuarua.

$$x\left(x-4\right)+7\left(x-4\right)$$

Whakatauwehea atu te kīanga pātahi x-4 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-4\right)\left(x+7\right)$$

Hei kimi otinga whārite, me whakaoti te x-4=0 me te x+7=0.

$$x=4 x=-7$$

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.

$$3x^{2}+9x+6=90$$

Me tango 90 mai i ngā taha e rua o te whārite.

$$3x^{2}+9x+6-90=90-90$$

Mā te tango i te 90 i a ia ake anō ka toe ko te 0.

$$3x^{2}+9x+6-90=0$$

Tango 90 mai i 6.

$$3x^{2}+9x-84=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 3 mō a, 9 mō b, me -84 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-9±\sqrt{9^{2}-4\times 3\left(-84\right)}}{2\times 3}$$

Pūrua 9.

$$x=\frac{-9±\sqrt{81-4\times 3\left(-84\right)}}{2\times 3}$$

Whakareatia -4 ki te 3.

$$x=\frac{-9±\sqrt{81-12\left(-84\right)}}{2\times 3}$$

Whakareatia -12 ki te -84.

$$x=\frac{-9±\sqrt{81+1008}}{2\times 3}$$

Tāpiri 81 ki te 1008.

$$x=\frac{-9±\sqrt{1089}}{2\times 3}$$

Tuhia te pūtakerua o te 1089.

$$x=\frac{-9±33}{2\times 3}$$

Whakareatia 2 ki te 3.

$$x=\frac{-9±33}{6}$$

Nā, me whakaoti te whārite x=\frac{-9±33}{6} ina he tāpiri te ±. Tāpiri -9 ki te 33.

$$x=\frac{24}{6}$$

Whakawehe 24 ki te 6.

$$x=4$$

Nā, me whakaoti te whārite x=\frac{-9±33}{6} ina he tango te ±. Tango 33 mai i -9.

$$x=-\frac{42}{6}$$

Whakawehe -42 ki te 6.

$$x=-7$$

Kua oti te whārite te whakatau.

$$x=4 x=-7$$

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.

$$3x^{2}+9x+6=90$$

Me tango 6 mai i ngā taha e rua o te whārite.

$$3x^{2}+9x+6-6=90-6$$

Mā te tango i te 6 i a ia ake anō ka toe ko te 0.

$$3x^{2}+9x=90-6$$

Tango 6 mai i 90.

$$3x^{2}+9x=84$$

Whakawehea ngā taha e rua ki te 3.

$$\frac{3x^{2}+9x}{3}=\frac{84}{3}$$

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

$$x^{2}+\frac{9}{3}x=\frac{84}{3}$$

Whakawehe 9 ki te 3.

$$x^{2}+3x=\frac{84}{3}$$

Whakawehe 84 ki te 3.

$$x^{2}+3x=28$$

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.

$$x^{2}+3x+\left(\frac{3}{2}\right)^{2}=28+\left(\frac{3}{2}\right)^{2}$$

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

$$x^{2}+3x+\frac{9}{4}=28+\frac{9}{4}$$

Tāpiri 28 ki te \frac{9}{4}.

$$x^{2}+3x+\frac{9}{4}=\frac{121}{4}$$

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}.

$$\left(x+\frac{3}{2}\right)^{2}=\frac{121}{4}$$

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

$$\sqrt{\left(x+\frac{3}{2}\right)^{2}}=\sqrt{\frac{121}{4}}$$

Whakarūnātia.

$$x+\frac{3}{2}=\frac{11}{2} x+\frac{3}{2}=-\frac{11}{2}$$

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

$$x=4 x=-7$$