Whakaoti i te 49.7=4x^2/(1-x)(2-x)

$$49.7= \frac{ 4 { x }^{ 2 } }{ (1-x)(2-x) }$$

Whakaoti mō x

$$x = \frac{\sqrt{406049} + 1491}{914} \approx 2.328467989$$

$$x=\frac{1491-\sqrt{406049}}{914}\approx 0.934114067$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara 1,2 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te \left(x-2\right)\left(x-1\right).

$$49.7\left(x-2\right)\left(x-1\right)=4x^{2}$$

Whakamahia te āhuatanga tohatoha hei whakarea te 49.7 ki te x-2.

$$\left(49.7x-99.4\right)\left(x-1\right)=4x^{2}$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 49.7x-99.4 ki te x-1 ka whakakotahi i ngā kupu rite.

$$49.7x^{2}-149.1x+99.4=4x^{2}$$

Tangohia te 4x^{2} mai i ngā taha e rua.

$$49.7x^{2}-149.1x+99.4-4x^{2}=0$$

Pahekotia te 49.7x^{2} me -4x^{2}, ka 45.7x^{2}.

$$45.7x^{2}-149.1x+99.4=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 45.7 mō a, -149.1 mō b, me 99.4 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-149.1\right)±\sqrt{\left(-149.1\right)^{2}-4\times 45.7\times 99.4}}{2\times 45.7}$$

Pūruatia -149.1 mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x=\frac{-\left(-149.1\right)±\sqrt{22230.81-4\times 45.7\times 99.4}}{2\times 45.7}$$

Whakareatia -4 ki te 45.7.

$$x=\frac{-\left(-149.1\right)±\sqrt{22230.81-182.8\times 99.4}}{2\times 45.7}$$

Whakareatia -182.8 ki te 99.4 mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=\frac{-\left(-149.1\right)±\sqrt{22230.81-18170.32}}{2\times 45.7}$$

Tāpiri 22230.81 ki te -18170.32 mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=\frac{-\left(-149.1\right)±\sqrt{4060.49}}{2\times 45.7}$$

Tuhia te pūtakerua o te 4060.49.

$$x=\frac{-\left(-149.1\right)±\frac{\sqrt{406049}}{10}}{2\times 45.7}$$

Ko te tauaro o -149.1 ko 149.1.

$$x=\frac{149.1±\frac{\sqrt{406049}}{10}}{2\times 45.7}$$

Whakareatia 2 ki te 45.7.

$$x=\frac{149.1±\frac{\sqrt{406049}}{10}}{91.4}$$

Nā, me whakaoti te whārite x=\frac{149.1±\frac{\sqrt{406049}}{10}}{91.4} ina he tāpiri te ±. Tāpiri 149.1 ki te \frac{\sqrt{406049}}{10}.

$$x=\frac{\sqrt{406049}+1491}{10\times 91.4}$$

Whakawehe \frac{1491+\sqrt{406049}}{10} ki te 91.4 mā te whakarea \frac{1491+\sqrt{406049}}{10} ki te tau huripoki o 91.4.

$$x=\frac{\sqrt{406049}+1491}{914}$$

Nā, me whakaoti te whārite x=\frac{149.1±\frac{\sqrt{406049}}{10}}{91.4} ina he tango te ±. Tango \frac{\sqrt{406049}}{10} mai i 149.1.

$$x=\frac{1491-\sqrt{406049}}{10\times 91.4}$$

Whakawehe \frac{1491-\sqrt{406049}}{10} ki te 91.4 mā te whakarea \frac{1491-\sqrt{406049}}{10} ki te tau huripoki o 91.4.

$$x=\frac{1491-\sqrt{406049}}{914}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{406049}+1491}{914} x=\frac{1491-\sqrt{406049}}{914}$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara 1,2 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te \left(x-2\right)\left(x-1\right).

$$49.7\left(x-2\right)\left(x-1\right)=4x^{2}$$

Whakamahia te āhuatanga tohatoha hei whakarea te 49.7 ki te x-2.

$$\left(49.7x-99.4\right)\left(x-1\right)=4x^{2}$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 49.7x-99.4 ki te x-1 ka whakakotahi i ngā kupu rite.

$$49.7x^{2}-149.1x+99.4=4x^{2}$$

Tangohia te 4x^{2} mai i ngā taha e rua.

$$49.7x^{2}-149.1x+99.4-4x^{2}=0$$

Pahekotia te 49.7x^{2} me -4x^{2}, ka 45.7x^{2}.

$$45.7x^{2}-149.1x+99.4=0$$

Tangohia te 99.4 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$45.7x^{2}-149.1x=-99.4$$

Whakawehea ngā taha e rua o te whārite ki te 45.7, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$\frac{45.7x^{2}-149.1x}{45.7}=-\frac{99.4}{45.7}$$

Mā te whakawehe ki te 45.7 ka wetekia te whakareanga ki te 45.7.

$$x^{2}+\left(-\frac{149.1}{45.7}\right)x=-\frac{99.4}{45.7}$$

Whakawehe -149.1 ki te 45.7 mā te whakarea -149.1 ki te tau huripoki o 45.7.

$$x^{2}-\frac{1491}{457}x=-\frac{99.4}{45.7}$$

Whakawehe -99.4 ki te 45.7 mā te whakarea -99.4 ki te tau huripoki o 45.7.

$$x^{2}-\frac{1491}{457}x=-\frac{994}{457}$$

Whakawehea te -\frac{1491}{457}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{1491}{914}. Nā, tāpiria te pūrua o te -\frac{1491}{914} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{1491}{457}x+\left(-\frac{1491}{914}\right)^{2}=-\frac{994}{457}+\left(-\frac{1491}{914}\right)^{2}$$

Pūruatia -\frac{1491}{914} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{1491}{457}x+\frac{2223081}{835396}=-\frac{994}{457}+\frac{2223081}{835396}$$

Tāpiri -\frac{994}{457} ki te \frac{2223081}{835396} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x^{2}-\frac{1491}{457}x+\frac{2223081}{835396}=\frac{406049}{835396}$$

Tauwehea x^{2}-\frac{1491}{457}x+\frac{2223081}{835396}. 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}.

$$\left(x-\frac{1491}{914}\right)^{2}=\frac{406049}{835396}$$

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

$$\sqrt{\left(x-\frac{1491}{914}\right)^{2}}=\sqrt{\frac{406049}{835396}}$$

Whakarūnātia.

$$x-\frac{1491}{914}=\frac{\sqrt{406049}}{914} x-\frac{1491}{914}=-\frac{\sqrt{406049}}{914}$$

Me tāpiri \frac{1491}{914} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{406049}+1491}{914} x=\frac{1491-\sqrt{406049}}{914}$$