Whakaoti i te 54(1+x)(1+x)=1215

$$54(1+x)(1+x)=1215$$

Whakaoti mō x

$$x=\frac{3\sqrt{10}}{2}-1\approx 3.74341649$$

$$x=-\frac{3\sqrt{10}}{2}-1\approx -5.74341649$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakareatia te 1+x ki te 1+x, ka \left(1+x\right)^{2}.

$$54\left(1+x\right)^{2}=1215$$

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

$$54\left(1+2x+x^{2}\right)=1215$$

Whakamahia te āhuatanga tohatoha hei whakarea te 54 ki te 1+2x+x^{2}.

$$54+108x+54x^{2}=1215$$

Tangohia te 1215 mai i ngā taha e rua.

$$54+108x+54x^{2}-1215=0$$

Tangohia te 1215 i te 54, ka -1161.

$$-1161+108x+54x^{2}=0$$

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.

$$54x^{2}+108x-1161=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 54 mō a, 108 mō b, me -1161 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-108±\sqrt{108^{2}-4\times 54\left(-1161\right)}}{2\times 54}$$

Pūrua 108.

$$x=\frac{-108±\sqrt{11664-4\times 54\left(-1161\right)}}{2\times 54}$$

Whakareatia -4 ki te 54.

$$x=\frac{-108±\sqrt{11664-216\left(-1161\right)}}{2\times 54}$$

Whakareatia -216 ki te -1161.

$$x=\frac{-108±\sqrt{11664+250776}}{2\times 54}$$

Tāpiri 11664 ki te 250776.

$$x=\frac{-108±\sqrt{262440}}{2\times 54}$$

Tuhia te pūtakerua o te 262440.

$$x=\frac{-108±162\sqrt{10}}{2\times 54}$$

Whakareatia 2 ki te 54.

$$x=\frac{-108±162\sqrt{10}}{108}$$

Nā, me whakaoti te whārite x=\frac{-108±162\sqrt{10}}{108} ina he tāpiri te ±. Tāpiri -108 ki te 162\sqrt{10}.

$$x=\frac{162\sqrt{10}-108}{108}$$

Whakawehe -108+162\sqrt{10} ki te 108.

$$x=\frac{3\sqrt{10}}{2}-1$$

Nā, me whakaoti te whārite x=\frac{-108±162\sqrt{10}}{108} ina he tango te ±. Tango 162\sqrt{10} mai i -108.

$$x=\frac{-162\sqrt{10}-108}{108}$$

Whakawehe -108-162\sqrt{10} ki te 108.

$$x=-\frac{3\sqrt{10}}{2}-1$$

Kua oti te whārite te whakatau.

$$x=\frac{3\sqrt{10}}{2}-1 x=-\frac{3\sqrt{10}}{2}-1$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakareatia te 1+x ki te 1+x, ka \left(1+x\right)^{2}.

$$54\left(1+x\right)^{2}=1215$$

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

$$54\left(1+2x+x^{2}\right)=1215$$

Whakamahia te āhuatanga tohatoha hei whakarea te 54 ki te 1+2x+x^{2}.

$$54+108x+54x^{2}=1215$$

Tangohia te 54 mai i ngā taha e rua.

$$108x+54x^{2}=1215-54$$

Tangohia te 54 i te 1215, ka 1161.

$$108x+54x^{2}=1161$$

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.

$$54x^{2}+108x=1161$$

Whakawehea ngā taha e rua ki te 54.

$$\frac{54x^{2}+108x}{54}=\frac{1161}{54}$$

Mā te whakawehe ki te 54 ka wetekia te whakareanga ki te 54.

$$x^{2}+\frac{108}{54}x=\frac{1161}{54}$$

Whakawehe 108 ki te 54.

$$x^{2}+2x=\frac{1161}{54}$$

Whakahekea te hautanga \frac{1161}{54} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 27.

$$x^{2}+2x=\frac{43}{2}$$

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.

$$x^{2}+2x+1^{2}=\frac{43}{2}+1^{2}$$

Pūrua 1.

$$x^{2}+2x+1=\frac{43}{2}+1$$

Tāpiri \frac{43}{2} ki te 1.

$$x^{2}+2x+1=\frac{45}{2}$$

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}.

$$\left(x+1\right)^{2}=\frac{45}{2}$$

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

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{\frac{45}{2}}$$

Whakarūnātia.

$$x+1=\frac{3\sqrt{10}}{2} x+1=-\frac{3\sqrt{10}}{2}$$

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

$$x=\frac{3\sqrt{10}}{2}-1 x=-\frac{3\sqrt{10}}{2}-1$$