Whakaoti i te x-(25%x-(40%)x)(10%x)=45

$$x - ( 25 \% x - ( 40 \% ) x ) ( 10 \% x ) = 45$$

Whakaoti mō x

$$x = \frac{10 \sqrt{370} - 100}{3} \approx 30.784613539$$

$$x=\frac{-10\sqrt{370}-100}{3}\approx -97.451280206$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakahekea te hautanga \frac{25}{100} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 25.

$$x-\left(\frac{1}{4}x-\frac{40}{100}x\right)\times \frac{10}{100}x=45$$

Whakahekea te hautanga \frac{40}{100} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 20.

$$x-\left(\frac{1}{4}x-\frac{2}{5}x\right)\times \frac{10}{100}x=45$$

Pahekotia te \frac{1}{4}x me -\frac{2}{5}x, ka -\frac{3}{20}x.

$$x-\left(-\frac{3}{20}x\times \frac{10}{100}x\right)=45$$

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

$$x-\left(-\frac{3}{20}x\times \frac{1}{10}x\right)=45$$

Me whakarea te -\frac{3}{20} ki te \frac{1}{10} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro.

$$x-\frac{-3}{20\times 10}xx=45$$

Mahia ngā whakarea i roto i te hautanga \frac{-3}{20\times 10}.

$$x-\frac{-3}{200}xx=45$$

Ka taea te hautanga \frac{-3}{200} te tuhi anō ko -\frac{3}{200} mā te tango i te tohu tōraro.

$$x-\left(-\frac{3}{200}xx\right)=45$$

Whakareatia te x ki te x, ka x^{2}.

$$x-\left(-\frac{3}{200}x^{2}\right)=45$$

Ko te tauaro o -\frac{3}{200}x^{2} ko \frac{3}{200}x^{2}.

$$x+\frac{3}{200}x^{2}=45$$

Tangohia te 45 mai i ngā taha e rua.

$$x+\frac{3}{200}x^{2}-45=0$$

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.

$$\frac{3}{200}x^{2}+x-45=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi \frac{3}{200} mō a, 1 mō b, me -45 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1±\sqrt{1^{2}-4\times \frac{3}{200}\left(-45\right)}}{2\times \frac{3}{200}}$$

Pūrua 1.

$$x=\frac{-1±\sqrt{1-4\times \frac{3}{200}\left(-45\right)}}{2\times \frac{3}{200}}$$

Whakareatia -4 ki te \frac{3}{200}.

$$x=\frac{-1±\sqrt{1-\frac{3}{50}\left(-45\right)}}{2\times \frac{3}{200}}$$

Whakareatia -\frac{3}{50} ki te -45.

$$x=\frac{-1±\sqrt{1+\frac{27}{10}}}{2\times \frac{3}{200}}$$

Tāpiri 1 ki te \frac{27}{10}.

$$x=\frac{-1±\sqrt{\frac{37}{10}}}{2\times \frac{3}{200}}$$

Tuhia te pūtakerua o te \frac{37}{10}.

$$x=\frac{-1±\frac{\sqrt{370}}{10}}{2\times \frac{3}{200}}$$

Whakareatia 2 ki te \frac{3}{200}.

$$x=\frac{-1±\frac{\sqrt{370}}{10}}{\frac{3}{100}}$$

Nā, me whakaoti te whārite x=\frac{-1±\frac{\sqrt{370}}{10}}{\frac{3}{100}} ina he tāpiri te ±. Tāpiri -1 ki te \frac{\sqrt{370}}{10}.

$$x=\frac{\frac{\sqrt{370}}{10}-1}{\frac{3}{100}}$$

Whakawehe -1+\frac{\sqrt{370}}{10} ki te \frac{3}{100} mā te whakarea -1+\frac{\sqrt{370}}{10} ki te tau huripoki o \frac{3}{100}.

$$x=\frac{10\sqrt{370}-100}{3}$$

Nā, me whakaoti te whārite x=\frac{-1±\frac{\sqrt{370}}{10}}{\frac{3}{100}} ina he tango te ±. Tango \frac{\sqrt{370}}{10} mai i -1.

$$x=\frac{-\frac{\sqrt{370}}{10}-1}{\frac{3}{100}}$$

Whakawehe -1-\frac{\sqrt{370}}{10} ki te \frac{3}{100} mā te whakarea -1-\frac{\sqrt{370}}{10} ki te tau huripoki o \frac{3}{100}.

$$x=\frac{-10\sqrt{370}-100}{3}$$

Kua oti te whārite te whakatau.

$$x=\frac{10\sqrt{370}-100}{3} x=\frac{-10\sqrt{370}-100}{3}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakahekea te hautanga \frac{25}{100} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 25.

$$x-\left(\frac{1}{4}x-\frac{40}{100}x\right)\times \frac{10}{100}x=45$$

Whakahekea te hautanga \frac{40}{100} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 20.

$$x-\left(\frac{1}{4}x-\frac{2}{5}x\right)\times \frac{10}{100}x=45$$

Pahekotia te \frac{1}{4}x me -\frac{2}{5}x, ka -\frac{3}{20}x.

$$x-\left(-\frac{3}{20}x\times \frac{10}{100}x\right)=45$$

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

$$x-\left(-\frac{3}{20}x\times \frac{1}{10}x\right)=45$$

Me whakarea te -\frac{3}{20} ki te \frac{1}{10} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro.

$$x-\frac{-3}{20\times 10}xx=45$$

Mahia ngā whakarea i roto i te hautanga \frac{-3}{20\times 10}.

$$x-\frac{-3}{200}xx=45$$

Ka taea te hautanga \frac{-3}{200} te tuhi anō ko -\frac{3}{200} mā te tango i te tohu tōraro.

$$x-\left(-\frac{3}{200}xx\right)=45$$

Whakareatia te x ki te x, ka x^{2}.

$$x-\left(-\frac{3}{200}x^{2}\right)=45$$

Ko te tauaro o -\frac{3}{200}x^{2} ko \frac{3}{200}x^{2}.

$$x+\frac{3}{200}x^{2}=45$$

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.

$$\frac{3}{200}x^{2}+x=45$$

Whakawehea ngā taha e rua o te whārite ki te \frac{3}{200}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$\frac{\frac{3}{200}x^{2}+x}{\frac{3}{200}}=\frac{45}{\frac{3}{200}}$$

Mā te whakawehe ki te \frac{3}{200} ka wetekia te whakareanga ki te \frac{3}{200}.

$$x^{2}+\frac{1}{\frac{3}{200}}x=\frac{45}{\frac{3}{200}}$$

Whakawehe 1 ki te \frac{3}{200} mā te whakarea 1 ki te tau huripoki o \frac{3}{200}.

$$x^{2}+\frac{200}{3}x=\frac{45}{\frac{3}{200}}$$

Whakawehe 45 ki te \frac{3}{200} mā te whakarea 45 ki te tau huripoki o \frac{3}{200}.

$$x^{2}+\frac{200}{3}x=3000$$

Whakawehea te \frac{200}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{100}{3}. Nā, tāpiria te pūrua o te \frac{100}{3} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{200}{3}x+\left(\frac{100}{3}\right)^{2}=3000+\left(\frac{100}{3}\right)^{2}$$

Pūruatia \frac{100}{3} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{200}{3}x+\frac{10000}{9}=3000+\frac{10000}{9}$$

Tāpiri 3000 ki te \frac{10000}{9}.

$$x^{2}+\frac{200}{3}x+\frac{10000}{9}=\frac{37000}{9}$$

Tauwehea x^{2}+\frac{200}{3}x+\frac{10000}{9}. 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}.

$$\left(x+\frac{100}{3}\right)^{2}=\frac{37000}{9}$$

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

$$\sqrt{\left(x+\frac{100}{3}\right)^{2}}=\sqrt{\frac{37000}{9}}$$

Whakarūnātia.

$$x+\frac{100}{3}=\frac{10\sqrt{370}}{3} x+\frac{100}{3}=-\frac{10\sqrt{370}}{3}$$

Me tango \frac{100}{3} mai i ngā taha e rua o te whārite.

$$x=\frac{10\sqrt{370}-100}{3} x=\frac{-10\sqrt{370}-100}{3}$$