Whakaoti i te 6^circ(1+x)^2=726

$$6 ^ { \circ } ( 1 + x ) ^ { 2 } = 726$$

Whakaoti mō x

$$x=10$$

$$x=-12$$

Ngā Upane Whakamahi Tauwehe

Whakawehea ngā taha e rua ki te 6.

$$\left(1+x\right)^{2}=\frac{726}{6}$$

Whakawehea te 726 ki te 6, kia riro ko 121.

$$\left(1+x\right)^{2}=121$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(1+x\right)^{2}.

$$1+2x+x^{2}=121$$

Tangohia te 121 mai i ngā taha e rua.

$$1+2x+x^{2}-121=0$$

Tangohia te 121 i te 1, ka -120.

$$-120+2x+x^{2}=0$$

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

$$x^{2}+2x-120=0$$

Hei whakaoti i te whārite, whakatauwehea te x^{2}+2x-120 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.

$$a+b=2 ab=-120$$

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 -120.

$$-1,120 -2,60 -3,40 -4,30 -5,24 -6,20 -8,15 -10,12$$

Tātaihia te tapeke mō ia takirua.

$$-1+120=119 -2+60=58 -3+40=37 -4+30=26 -5+24=19 -6+20=14 -8+15=7 -10+12=2$$

Ko te otinga te takirua ka hoatu i te tapeke 2.

$$a=-10 b=12$$

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

$$\left(x-10\right)\left(x+12\right)$$

Hei kimi otinga whārite, me whakaoti te x-10=0 me te x+12=0.

$$x=10 x=-12$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakawehea ngā taha e rua ki te 6.

$$\left(1+x\right)^{2}=\frac{726}{6}$$

Whakawehea te 726 ki te 6, kia riro ko 121.

$$\left(1+x\right)^{2}=121$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(1+x\right)^{2}.

$$1+2x+x^{2}=121$$

Tangohia te 121 mai i ngā taha e rua.

$$1+2x+x^{2}-121=0$$

Tangohia te 121 i te 1, ka -120.

$$-120+2x+x^{2}=0$$

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

$$x^{2}+2x-120=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx-120. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=2 ab=1\left(-120\right)=-120$$

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 -120.

$$-1,120 -2,60 -3,40 -4,30 -5,24 -6,20 -8,15 -10,12$$

Tātaihia te tapeke mō ia takirua.

$$-1+120=119 -2+60=58 -3+40=37 -4+30=26 -5+24=19 -6+20=14 -8+15=7 -10+12=2$$

Ko te otinga te takirua ka hoatu i te tapeke 2.

$$a=-10 b=12$$

Tuhia anō te x^{2}+2x-120 hei \left(x^{2}-10x\right)+\left(12x-120\right).

$$\left(x^{2}-10x\right)+\left(12x-120\right)$$

Tauwehea te x i te tuatahi me te 12 i te rōpū tuarua.

$$x\left(x-10\right)+12\left(x-10\right)$$

Whakatauwehea atu te kīanga pātahi x-10 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-10\right)\left(x+12\right)$$

Hei kimi otinga whārite, me whakaoti te x-10=0 me te x+12=0.

$$x=10 x=-12$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakawehea ngā taha e rua ki te 6.

$$\left(1+x\right)^{2}=\frac{726}{6}$$

Whakawehea te 726 ki te 6, kia riro ko 121.

$$\left(1+x\right)^{2}=121$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(1+x\right)^{2}.

$$1+2x+x^{2}=121$$

Tangohia te 121 mai i ngā taha e rua.

$$1+2x+x^{2}-121=0$$

Tangohia te 121 i te 1, ka -120.

$$-120+2x+x^{2}=0$$

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.

$$x^{2}+2x-120=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 2 mō b, me -120 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Pūrua 2.

$$x=\frac{-2±\sqrt{4-4\left(-120\right)}}{2}$$

Whakareatia -4 ki te -120.

$$x=\frac{-2±\sqrt{4+480}}{2}$$

Tāpiri 4 ki te 480.

$$x=\frac{-2±\sqrt{484}}{2}$$

Tuhia te pūtakerua o te 484.

$$x=\frac{-2±22}{2}$$

Nā, me whakaoti te whārite x=\frac{-2±22}{2} ina he tāpiri te ±. Tāpiri -2 ki te 22.

$$x=\frac{20}{2}$$

Whakawehe 20 ki te 2.

$$x=10$$

Nā, me whakaoti te whārite x=\frac{-2±22}{2} ina he tango te ±. Tango 22 mai i -2.

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

Whakawehe -24 ki te 2.

$$x=-12$$

Kua oti te whārite te whakatau.

$$x=10 x=-12$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakawehea ngā taha e rua ki te 6.

$$\left(1+x\right)^{2}=\frac{726}{6}$$

Whakawehea te 726 ki te 6, kia riro ko 121.

$$\left(1+x\right)^{2}=121$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(1+x\right)^{2}.

$$1+2x+x^{2}=121$$

Tangohia te 1 mai i ngā taha e rua.

$$2x+x^{2}=121-1$$

Tangohia te 1 i te 121, ka 120.

$$2x+x^{2}=120$$

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.

$$x^{2}+2x=120$$

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

$$x^{2}+2x+1^{2}=120+1^{2}$$

Pūrua 1.

$$x^{2}+2x+1=120+1$$

Tāpiri 120 ki te 1.

$$x^{2}+2x+1=121$$

Tauwehea x^{2}+2x+1. 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}.

$$\left(x+1\right)^{2}=121$$

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

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{121}$$

Whakarūnātia.

$$x+1=11 x+1=-11$$

Me tango 1 mai i ngā taha e rua o te whārite.

$$x=10 x=-12$$