Whakaoti i te 2200/100-x+15=22*100/67-x

$$\frac{ 2200 }{ 100-x } +15= \frac{ 22 \times 100 }{ 67-x }$$

Whakaoti mō x

$$x=12$$

$$x=155$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara 67,100 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te \left(x-100\right)\left(x-67\right), arā, te tauraro pātahi he tino iti rawa te kitea o 100-x,67-x.

$$\left(67-x\right)\times 2200+\left(x-100\right)\left(x-67\right)\times 15=\left(100-x\right)\times 22\times 100$$

Whakamahia te āhuatanga tohatoha hei whakarea te 67-x ki te 2200.

$$147400-2200x+\left(x-100\right)\left(x-67\right)\times 15=\left(100-x\right)\times 22\times 100$$

Whakamahia te āhuatanga tuaritanga hei whakarea te x-100 ki te x-67 ka whakakotahi i ngā kupu rite.

$$147400-2200x+\left(x^{2}-167x+6700\right)\times 15=\left(100-x\right)\times 22\times 100$$

Whakamahia te āhuatanga tohatoha hei whakarea te x^{2}-167x+6700 ki te 15.

$$147400-2200x+15x^{2}-2505x+100500=\left(100-x\right)\times 22\times 100$$

Pahekotia te -2200x me -2505x, ka -4705x.

$$147400-4705x+15x^{2}+100500=\left(100-x\right)\times 22\times 100$$

Tāpirihia te 147400 ki te 100500, ka 247900.

$$247900-4705x+15x^{2}=\left(100-x\right)\times 22\times 100$$

Whakareatia te 22 ki te 100, ka 2200.

$$247900-4705x+15x^{2}=\left(100-x\right)\times 2200$$

Whakamahia te āhuatanga tohatoha hei whakarea te 100-x ki te 2200.

$$247900-4705x+15x^{2}=220000-2200x$$

Tangohia te 220000 mai i ngā taha e rua.

$$247900-4705x+15x^{2}-220000=-2200x$$

Tangohia te 220000 i te 247900, ka 27900.

$$27900-4705x+15x^{2}=-2200x$$

Me tāpiri te 2200x ki ngā taha e rua.

$$27900-4705x+15x^{2}+2200x=0$$

Pahekotia te -4705x me 2200x, ka -2505x.

$$27900-2505x+15x^{2}=0$$

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.

$$15x^{2}-2505x+27900=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 15 mō a, -2505 mō b, me 27900 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-2505\right)±\sqrt{\left(-2505\right)^{2}-4\times 15\times 27900}}{2\times 15}$$

Pūrua -2505.

$$x=\frac{-\left(-2505\right)±\sqrt{6275025-4\times 15\times 27900}}{2\times 15}$$

Whakareatia -4 ki te 15.

$$x=\frac{-\left(-2505\right)±\sqrt{6275025-60\times 27900}}{2\times 15}$$

Whakareatia -60 ki te 27900.

$$x=\frac{-\left(-2505\right)±\sqrt{6275025-1674000}}{2\times 15}$$

Tāpiri 6275025 ki te -1674000.

$$x=\frac{-\left(-2505\right)±\sqrt{4601025}}{2\times 15}$$

Tuhia te pūtakerua o te 4601025.

$$x=\frac{-\left(-2505\right)±2145}{2\times 15}$$

Ko te tauaro o -2505 ko 2505.

$$x=\frac{2505±2145}{2\times 15}$$

Whakareatia 2 ki te 15.

$$x=\frac{2505±2145}{30}$$

Nā, me whakaoti te whārite x=\frac{2505±2145}{30} ina he tāpiri te ±. Tāpiri 2505 ki te 2145.

$$x=\frac{4650}{30}$$

Whakawehe 4650 ki te 30.

$$x=155$$

Nā, me whakaoti te whārite x=\frac{2505±2145}{30} ina he tango te ±. Tango 2145 mai i 2505.

$$x=\frac{360}{30}$$

Whakawehe 360 ki te 30.

$$x=12$$

Kua oti te whārite te whakatau.

$$x=155 x=12$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara 67,100 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te \left(x-100\right)\left(x-67\right), arā, te tauraro pātahi he tino iti rawa te kitea o 100-x,67-x.

$$\left(67-x\right)\times 2200+\left(x-100\right)\left(x-67\right)\times 15=\left(100-x\right)\times 22\times 100$$

Whakamahia te āhuatanga tohatoha hei whakarea te 67-x ki te 2200.

$$147400-2200x+\left(x-100\right)\left(x-67\right)\times 15=\left(100-x\right)\times 22\times 100$$

Whakamahia te āhuatanga tuaritanga hei whakarea te x-100 ki te x-67 ka whakakotahi i ngā kupu rite.

$$147400-2200x+\left(x^{2}-167x+6700\right)\times 15=\left(100-x\right)\times 22\times 100$$

Whakamahia te āhuatanga tohatoha hei whakarea te x^{2}-167x+6700 ki te 15.

$$147400-2200x+15x^{2}-2505x+100500=\left(100-x\right)\times 22\times 100$$

Pahekotia te -2200x me -2505x, ka -4705x.

$$147400-4705x+15x^{2}+100500=\left(100-x\right)\times 22\times 100$$

Tāpirihia te 147400 ki te 100500, ka 247900.

$$247900-4705x+15x^{2}=\left(100-x\right)\times 22\times 100$$

Whakareatia te 22 ki te 100, ka 2200.

$$247900-4705x+15x^{2}=\left(100-x\right)\times 2200$$

Whakamahia te āhuatanga tohatoha hei whakarea te 100-x ki te 2200.

$$247900-4705x+15x^{2}=220000-2200x$$

Me tāpiri te 2200x ki ngā taha e rua.

$$247900-4705x+15x^{2}+2200x=220000$$

Pahekotia te -4705x me 2200x, ka -2505x.

$$247900-2505x+15x^{2}=220000$$

Tangohia te 247900 mai i ngā taha e rua.

$$-2505x+15x^{2}=220000-247900$$

Tangohia te 247900 i te 220000, ka -27900.

$$-2505x+15x^{2}=-27900$$

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.

$$15x^{2}-2505x=-27900$$

Whakawehea ngā taha e rua ki te 15.

$$\frac{15x^{2}-2505x}{15}=-\frac{27900}{15}$$

Mā te whakawehe ki te 15 ka wetekia te whakareanga ki te 15.

$$x^{2}+\left(-\frac{2505}{15}\right)x=-\frac{27900}{15}$$

Whakawehe -2505 ki te 15.

$$x^{2}-167x=-\frac{27900}{15}$$

Whakawehe -27900 ki te 15.

$$x^{2}-167x=-1860$$

Whakawehea te -167, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{167}{2}. Nā, tāpiria te pūrua o te -\frac{167}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-167x+\left(-\frac{167}{2}\right)^{2}=-1860+\left(-\frac{167}{2}\right)^{2}$$

Pūruatia -\frac{167}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-167x+\frac{27889}{4}=-1860+\frac{27889}{4}$$

Tāpiri -1860 ki te \frac{27889}{4}.

$$x^{2}-167x+\frac{27889}{4}=\frac{20449}{4}$$

Tauwehea x^{2}-167x+\frac{27889}{4}. 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}.

$$\left(x-\frac{167}{2}\right)^{2}=\frac{20449}{4}$$

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

$$\sqrt{\left(x-\frac{167}{2}\right)^{2}}=\sqrt{\frac{20449}{4}}$$

Whakarūnātia.

$$x-\frac{167}{2}=\frac{143}{2} x-\frac{167}{2}=-\frac{143}{2}$$

Me tāpiri \frac{167}{2} ki ngā taha e rua o te whārite.

$$x=155 x=12$$