Whakaoti i te 54(1+x)(1+x)=121.5

$$54(1+x)(1+x)=121.5$$

Whakaoti mō x

$$x=0.5$$

$$x=-2.5$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

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

$$54\left(1+x\right)^{2}=121.5$$

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

$$54\left(1+2x+x^{2}\right)=121.5$$

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

$$54+108x+54x^{2}=121.5$$

Tangohia te 121.5 mai i ngā taha e rua.

$$54+108x+54x^{2}-121.5=0$$

Tangohia te 121.5 i te 54, ka -67.5.

$$-67.5+108x+54x^{2}=0$$

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.

$$54x^{2}+108x-67.5=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 54 mō a, 108 mō b, me -67.5 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-108±\sqrt{108^{2}-4\times 54\left(-67.5\right)}}{2\times 54}$$

Pūrua 108.

$$x=\frac{-108±\sqrt{11664-4\times 54\left(-67.5\right)}}{2\times 54}$$

Whakareatia -4 ki te 54.

$$x=\frac{-108±\sqrt{11664-216\left(-67.5\right)}}{2\times 54}$$

Whakareatia -216 ki te -67.5.

$$x=\frac{-108±\sqrt{11664+14580}}{2\times 54}$$

Tāpiri 11664 ki te 14580.

$$x=\frac{-108±\sqrt{26244}}{2\times 54}$$

Tuhia te pūtakerua o te 26244.

$$x=\frac{-108±162}{2\times 54}$$

Whakareatia 2 ki te 54.

$$x=\frac{-108±162}{108}$$

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

$$x=\frac{54}{108}$$

Whakahekea te hautanga \frac{54}{108} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 54.

$$x=\frac{1}{2}$$

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

$$x=-\frac{270}{108}$$

Whakahekea te hautanga \frac{-270}{108} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 54.

$$x=-\frac{5}{2}$$

Kua oti te whārite te whakatau.

$$x=\frac{1}{2} x=-\frac{5}{2}$$

Ngā Upane mō te Whakaoti i te Pūrua

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

$$54\left(1+x\right)^{2}=121.5$$

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

$$54\left(1+2x+x^{2}\right)=121.5$$

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

$$54+108x+54x^{2}=121.5$$

Tangohia te 54 mai i ngā taha e rua.

$$108x+54x^{2}=121.5-54$$

Tangohia te 54 i te 121.5, ka 67.5.

$$108x+54x^{2}=67.5$$

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.

$$54x^{2}+108x=67.5$$

Whakawehea ngā taha e rua ki te 54.

$$\frac{54x^{2}+108x}{54}=\frac{67.5}{54}$$

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

$$x^{2}+\frac{108}{54}x=\frac{67.5}{54}$$

Whakawehe 108 ki te 54.

$$x^{2}+2x=\frac{67.5}{54}$$

Whakawehe 67.5 ki te 54.

$$x^{2}+2x=1.25$$

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.

$$x^{2}+2x+1^{2}=1.25+1^{2}$$

Pūrua 1.

$$x^{2}+2x+1=1.25+1$$

Tāpiri 1.25 ki te 1.

$$x^{2}+2x+1=2.25$$

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}.

$$\left(x+1\right)^{2}=2.25$$

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

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{2.25}$$

Whakarūnātia.

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

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

$$x=\frac{1}{2} x=-\frac{5}{2}$$