Whakaoti i te 50(1-10%)(1+x)^2=148

$$50 ( 1 - 10 \% ) ( 1 + x ) ^ { 2 } = 148$$

Whakaoti mō x

$$x=\frac{2\sqrt{185}}{15}-1\approx 0.813529401$$

$$x=-\frac{2\sqrt{185}}{15}-1\approx -2.813529401$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakahekea te hautanga \frac{10}{100} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 10.

$$50\left(1-\frac{1}{10}\right)\left(1+x\right)^{2}=148$$

Tangohia te \frac{1}{10} i te 1, ka \frac{9}{10}.

$$50\times \frac{9}{10}\left(1+x\right)^{2}=148$$

Whakareatia te 50 ki te \frac{9}{10}, ka 45.

$$45\left(1+x\right)^{2}=148$$

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

$$45\left(1+2x+x^{2}\right)=148$$

Whakamahia te āhuatanga tohatoha hei whakarea te 45 ki te 1+2x+x^{2}.

$$45+90x+45x^{2}=148$$

Tangohia te 148 mai i ngā taha e rua.

$$45+90x+45x^{2}-148=0$$

Tangohia te 148 i te 45, ka -103.

$$-103+90x+45x^{2}=0$$

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.

$$45x^{2}+90x-103=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 45 mō a, 90 mō b, me -103 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-90±\sqrt{90^{2}-4\times 45\left(-103\right)}}{2\times 45}$$

Pūrua 90.

$$x=\frac{-90±\sqrt{8100-4\times 45\left(-103\right)}}{2\times 45}$$

Whakareatia -4 ki te 45.

$$x=\frac{-90±\sqrt{8100-180\left(-103\right)}}{2\times 45}$$

Whakareatia -180 ki te -103.

$$x=\frac{-90±\sqrt{8100+18540}}{2\times 45}$$

Tāpiri 8100 ki te 18540.

$$x=\frac{-90±\sqrt{26640}}{2\times 45}$$

Tuhia te pūtakerua o te 26640.

$$x=\frac{-90±12\sqrt{185}}{2\times 45}$$

Whakareatia 2 ki te 45.

$$x=\frac{-90±12\sqrt{185}}{90}$$

Nā, me whakaoti te whārite x=\frac{-90±12\sqrt{185}}{90} ina he tāpiri te ±. Tāpiri -90 ki te 12\sqrt{185}.

$$x=\frac{12\sqrt{185}-90}{90}$$

Whakawehe -90+12\sqrt{185} ki te 90.

$$x=\frac{2\sqrt{185}}{15}-1$$

Nā, me whakaoti te whārite x=\frac{-90±12\sqrt{185}}{90} ina he tango te ±. Tango 12\sqrt{185} mai i -90.

$$x=\frac{-12\sqrt{185}-90}{90}$$

Whakawehe -90-12\sqrt{185} ki te 90.

$$x=-\frac{2\sqrt{185}}{15}-1$$

Kua oti te whārite te whakatau.

$$x=\frac{2\sqrt{185}}{15}-1 x=-\frac{2\sqrt{185}}{15}-1$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakahekea te hautanga \frac{10}{100} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 10.

$$50\left(1-\frac{1}{10}\right)\left(1+x\right)^{2}=148$$

Tangohia te \frac{1}{10} i te 1, ka \frac{9}{10}.

$$50\times \frac{9}{10}\left(1+x\right)^{2}=148$$

Whakareatia te 50 ki te \frac{9}{10}, ka 45.

$$45\left(1+x\right)^{2}=148$$

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

$$45\left(1+2x+x^{2}\right)=148$$

Whakamahia te āhuatanga tohatoha hei whakarea te 45 ki te 1+2x+x^{2}.

$$45+90x+45x^{2}=148$$

Tangohia te 45 mai i ngā taha e rua.

$$90x+45x^{2}=148-45$$

Tangohia te 45 i te 148, ka 103.

$$90x+45x^{2}=103$$

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.

$$45x^{2}+90x=103$$

Whakawehea ngā taha e rua ki te 45.

$$\frac{45x^{2}+90x}{45}=\frac{103}{45}$$

Mā te whakawehe ki te 45 ka wetekia te whakareanga ki te 45.

$$x^{2}+\frac{90}{45}x=\frac{103}{45}$$

Whakawehe 90 ki te 45.

$$x^{2}+2x=\frac{103}{45}$$

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.

$$x^{2}+2x+1^{2}=\frac{103}{45}+1^{2}$$

Pūrua 1.

$$x^{2}+2x+1=\frac{103}{45}+1$$

Tāpiri \frac{103}{45} ki te 1.

$$x^{2}+2x+1=\frac{148}{45}$$

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}.

$$\left(x+1\right)^{2}=\frac{148}{45}$$

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

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{\frac{148}{45}}$$

Whakarūnātia.

$$x+1=\frac{2\sqrt{185}}{15} x+1=-\frac{2\sqrt{185}}{15}$$

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

$$x=\frac{2\sqrt{185}}{15}-1 x=-\frac{2\sqrt{185}}{15}-1$$