Whakaoti i te 99/100=9n^2-9n+2/9n^2-1

$$\frac { 99 } { 100 } = \frac { 9 n ^ { 2 } - 9 n + 2 } { 9 n ^ { 2 } - 1 }$$

Whakaoti mō n

$$n = \frac{299}{3} = 99\frac{2}{3} \approx 99.666666667$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe n ki tētahi o ngā uara -\frac{1}{3},\frac{1}{3} nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 100\left(3n-1\right)\left(3n+1\right), arā, te tauraro pātahi he tino iti rawa te kitea o 100,9n^{2}-1.

$$99\left(3n-1\right)\left(3n+1\right)=100\left(9n^{2}-9n+2\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 99 ki te 3n-1.

$$\left(297n-99\right)\left(3n+1\right)=100\left(9n^{2}-9n+2\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 297n-99 ki te 3n+1 ka whakakotahi i ngā kupu rite.

$$891n^{2}-99=100\left(9n^{2}-9n+2\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 100 ki te 9n^{2}-9n+2.

$$891n^{2}-99=900n^{2}-900n+200$$

Tangohia te 900n^{2} mai i ngā taha e rua.

$$891n^{2}-99-900n^{2}=-900n+200$$

Pahekotia te 891n^{2} me -900n^{2}, ka -9n^{2}.

$$-9n^{2}-99=-900n+200$$

Me tāpiri te 900n ki ngā taha e rua.

$$-9n^{2}-99+900n=200$$

Tangohia te 200 mai i ngā taha e rua.

$$-9n^{2}-99+900n-200=0$$

Tangohia te 200 i te -99, ka -299.

$$-9n^{2}-299+900n=0$$

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.

$$-9n^{2}+900n-299=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -9 mō a, 900 mō b, me -299 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-900±\sqrt{900^{2}-4\left(-9\right)\left(-299\right)}}{2\left(-9\right)}$$

Pūrua 900.

$$n=\frac{-900±\sqrt{810000-4\left(-9\right)\left(-299\right)}}{2\left(-9\right)}$$

Whakareatia -4 ki te -9.

$$n=\frac{-900±\sqrt{810000+36\left(-299\right)}}{2\left(-9\right)}$$

Whakareatia 36 ki te -299.

$$n=\frac{-900±\sqrt{810000-10764}}{2\left(-9\right)}$$

Tāpiri 810000 ki te -10764.

$$n=\frac{-900±\sqrt{799236}}{2\left(-9\right)}$$

Tuhia te pūtakerua o te 799236.

$$n=\frac{-900±894}{2\left(-9\right)}$$

Whakareatia 2 ki te -9.

$$n=\frac{-900±894}{-18}$$

Nā, me whakaoti te whārite n=\frac{-900±894}{-18} ina he tāpiri te ±. Tāpiri -900 ki te 894.

$$n=-\frac{6}{-18}$$

Whakahekea te hautanga \frac{-6}{-18} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 6.

$$n=\frac{1}{3}$$

Nā, me whakaoti te whārite n=\frac{-900±894}{-18} ina he tango te ±. Tango 894 mai i -900.

$$n=-\frac{1794}{-18}$$

Whakahekea te hautanga \frac{-1794}{-18} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 6.

$$n=\frac{299}{3}$$

Kua oti te whārite te whakatau.

$$n=\frac{1}{3} n=\frac{299}{3}$$

Tē taea kia ōrite te tāupe n ki \frac{1}{3}.

$$n=\frac{299}{3}$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe n ki tētahi o ngā uara -\frac{1}{3},\frac{1}{3} nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 100\left(3n-1\right)\left(3n+1\right), arā, te tauraro pātahi he tino iti rawa te kitea o 100,9n^{2}-1.

$$99\left(3n-1\right)\left(3n+1\right)=100\left(9n^{2}-9n+2\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 99 ki te 3n-1.

$$\left(297n-99\right)\left(3n+1\right)=100\left(9n^{2}-9n+2\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 297n-99 ki te 3n+1 ka whakakotahi i ngā kupu rite.

$$891n^{2}-99=100\left(9n^{2}-9n+2\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 100 ki te 9n^{2}-9n+2.

$$891n^{2}-99=900n^{2}-900n+200$$

Tangohia te 900n^{2} mai i ngā taha e rua.

$$891n^{2}-99-900n^{2}=-900n+200$$

Pahekotia te 891n^{2} me -900n^{2}, ka -9n^{2}.

$$-9n^{2}-99=-900n+200$$

Me tāpiri te 900n ki ngā taha e rua.

$$-9n^{2}-99+900n=200$$

Me tāpiri te 99 ki ngā taha e rua.

$$-9n^{2}+900n=200+99$$

Tāpirihia te 200 ki te 99, ka 299.

$$-9n^{2}+900n=299$$

Whakawehea ngā taha e rua ki te -9.

$$\frac{-9n^{2}+900n}{-9}=\frac{299}{-9}$$

Mā te whakawehe ki te -9 ka wetekia te whakareanga ki te -9.

$$n^{2}+\frac{900}{-9}n=\frac{299}{-9}$$

Whakawehe 900 ki te -9.

$$n^{2}-100n=\frac{299}{-9}$$

Whakawehe 299 ki te -9.

$$n^{2}-100n=-\frac{299}{9}$$

Whakawehea te -100, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -50. Nā, tāpiria te pūrua o te -50 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$n^{2}-100n+\left(-50\right)^{2}=-\frac{299}{9}+\left(-50\right)^{2}$$

Pūrua -50.

$$n^{2}-100n+2500=-\frac{299}{9}+2500$$

Tāpiri -\frac{299}{9} ki te 2500.

$$n^{2}-100n+2500=\frac{22201}{9}$$

Tauwehea n^{2}-100n+2500. 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}.

$$\left(n-50\right)^{2}=\frac{22201}{9}$$

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

$$\sqrt{\left(n-50\right)^{2}}=\sqrt{\frac{22201}{9}}$$

Whakarūnātia.

$$n-50=\frac{149}{3} n-50=-\frac{149}{3}$$

Me tāpiri 50 ki ngā taha e rua o te whārite.

$$n=\frac{299}{3} n=\frac{1}{3}$$

Tē taea kia ōrite te tāupe n ki \frac{1}{3}.

$$n=\frac{299}{3}$$