Whakaoti i te 4x+102=-20x(3-6x)

$$4 x + 102 = - 20 x ( 3 - 6 x )$$

Whakaoti mō x

$$x=\frac{\sqrt{829}}{30}+\frac{4}{15}\approx 1.226412003$$

$$x=-\frac{\sqrt{829}}{30}+\frac{4}{15}\approx -0.69307867$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tohatoha hei whakarea te -20x ki te 3-6x.

$$4x+102=-60x+120x^{2}$$

Me tāpiri te 60x ki ngā taha e rua.

$$4x+102+60x=120x^{2}$$

Pahekotia te 4x me 60x, ka 64x.

$$64x+102=120x^{2}$$

Tangohia te 120x^{2} mai i ngā taha e rua.

$$64x+102-120x^{2}=0$$

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.

$$-120x^{2}+64x+102=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -120 mō a, 64 mō b, me 102 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-64±\sqrt{64^{2}-4\left(-120\right)\times 102}}{2\left(-120\right)}$$

Pūrua 64.

$$x=\frac{-64±\sqrt{4096-4\left(-120\right)\times 102}}{2\left(-120\right)}$$

Whakareatia -4 ki te -120.

$$x=\frac{-64±\sqrt{4096+480\times 102}}{2\left(-120\right)}$$

Whakareatia 480 ki te 102.

$$x=\frac{-64±\sqrt{4096+48960}}{2\left(-120\right)}$$

Tāpiri 4096 ki te 48960.

$$x=\frac{-64±\sqrt{53056}}{2\left(-120\right)}$$

Tuhia te pūtakerua o te 53056.

$$x=\frac{-64±8\sqrt{829}}{2\left(-120\right)}$$

Whakareatia 2 ki te -120.

$$x=\frac{-64±8\sqrt{829}}{-240}$$

Nā, me whakaoti te whārite x=\frac{-64±8\sqrt{829}}{-240} ina he tāpiri te ±. Tāpiri -64 ki te 8\sqrt{829}.

$$x=\frac{8\sqrt{829}-64}{-240}$$

Whakawehe -64+8\sqrt{829} ki te -240.

$$x=-\frac{\sqrt{829}}{30}+\frac{4}{15}$$

Nā, me whakaoti te whārite x=\frac{-64±8\sqrt{829}}{-240} ina he tango te ±. Tango 8\sqrt{829} mai i -64.

$$x=\frac{-8\sqrt{829}-64}{-240}$$

Whakawehe -64-8\sqrt{829} ki te -240.

$$x=\frac{\sqrt{829}}{30}+\frac{4}{15}$$

Kua oti te whārite te whakatau.

$$x=-\frac{\sqrt{829}}{30}+\frac{4}{15} x=\frac{\sqrt{829}}{30}+\frac{4}{15}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tohatoha hei whakarea te -20x ki te 3-6x.

$$4x+102=-60x+120x^{2}$$

Me tāpiri te 60x ki ngā taha e rua.

$$4x+102+60x=120x^{2}$$

Pahekotia te 4x me 60x, ka 64x.

$$64x+102=120x^{2}$$

Tangohia te 120x^{2} mai i ngā taha e rua.

$$64x+102-120x^{2}=0$$

Tangohia te 102 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$64x-120x^{2}=-102$$

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.

$$-120x^{2}+64x=-102$$

Whakawehea ngā taha e rua ki te -120.

$$\frac{-120x^{2}+64x}{-120}=-\frac{102}{-120}$$

Mā te whakawehe ki te -120 ka wetekia te whakareanga ki te -120.

$$x^{2}+\frac{64}{-120}x=-\frac{102}{-120}$$

Whakahekea te hautanga \frac{64}{-120} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 8.

$$x^{2}-\frac{8}{15}x=-\frac{102}{-120}$$

Whakahekea te hautanga \frac{-102}{-120} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 6.

$$x^{2}-\frac{8}{15}x=\frac{17}{20}$$

Whakawehea te -\frac{8}{15}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{4}{15}. Nā, tāpiria te pūrua o te -\frac{4}{15} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{8}{15}x+\left(-\frac{4}{15}\right)^{2}=\frac{17}{20}+\left(-\frac{4}{15}\right)^{2}$$

Pūruatia -\frac{4}{15} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{8}{15}x+\frac{16}{225}=\frac{17}{20}+\frac{16}{225}$$

Tāpiri \frac{17}{20} ki te \frac{16}{225} 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.

$$x^{2}-\frac{8}{15}x+\frac{16}{225}=\frac{829}{900}$$

Tauwehea x^{2}-\frac{8}{15}x+\frac{16}{225}. 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}.

$$\left(x-\frac{4}{15}\right)^{2}=\frac{829}{900}$$

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

$$\sqrt{\left(x-\frac{4}{15}\right)^{2}}=\sqrt{\frac{829}{900}}$$

Whakarūnātia.

$$x-\frac{4}{15}=\frac{\sqrt{829}}{30} x-\frac{4}{15}=-\frac{\sqrt{829}}{30}$$

Me tāpiri \frac{4}{15} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{829}}{30}+\frac{4}{15} x=-\frac{\sqrt{829}}{30}+\frac{4}{15}$$