Whakaoti i te 3*(2x-10)2(3x-30)=-5(3x+100)

$$3 \times (2x-10)2(3x-30)=-5(3x+100)$$

Whakaoti mō x (complex solution)

$$x=\frac{175+5\sqrt{247}i}{24}\approx 7.291666667+3.274215343i$$

$$x=\frac{-5\sqrt{247}i+175}{24}\approx 7.291666667-3.274215343i$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakareatia te 3 ki te 2, ka 6.

$$6\left(2x-10\right)\left(3x-30\right)=-5\left(3x+100\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 6 ki te 2x-10.

$$\left(12x-60\right)\left(3x-30\right)=-5\left(3x+100\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 12x-60 ki te 3x-30 ka whakakotahi i ngā kupu rite.

$$36x^{2}-540x+1800=-5\left(3x+100\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te -5 ki te 3x+100.

$$36x^{2}-540x+1800=-15x-500$$

Me tāpiri te 15x ki ngā taha e rua.

$$36x^{2}-540x+1800+15x=-500$$

Pahekotia te -540x me 15x, ka -525x.

$$36x^{2}-525x+1800=-500$$

Me tāpiri te 500 ki ngā taha e rua.

$$36x^{2}-525x+1800+500=0$$

Tāpirihia te 1800 ki te 500, ka 2300.

$$36x^{2}-525x+2300=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 36 mō a, -525 mō b, me 2300 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-525\right)±\sqrt{\left(-525\right)^{2}-4\times 36\times 2300}}{2\times 36}$$

Pūrua -525.

$$x=\frac{-\left(-525\right)±\sqrt{275625-4\times 36\times 2300}}{2\times 36}$$

Whakareatia -4 ki te 36.

$$x=\frac{-\left(-525\right)±\sqrt{275625-144\times 2300}}{2\times 36}$$

Whakareatia -144 ki te 2300.

$$x=\frac{-\left(-525\right)±\sqrt{275625-331200}}{2\times 36}$$

Tāpiri 275625 ki te -331200.

$$x=\frac{-\left(-525\right)±\sqrt{-55575}}{2\times 36}$$

Tuhia te pūtakerua o te -55575.

$$x=\frac{-\left(-525\right)±15\sqrt{247}i}{2\times 36}$$

Ko te tauaro o -525 ko 525.

$$x=\frac{525±15\sqrt{247}i}{2\times 36}$$

Whakareatia 2 ki te 36.

$$x=\frac{525±15\sqrt{247}i}{72}$$

Nā, me whakaoti te whārite x=\frac{525±15\sqrt{247}i}{72} ina he tāpiri te ±. Tāpiri 525 ki te 15i\sqrt{247}.

$$x=\frac{525+15\sqrt{247}i}{72}$$

Whakawehe 525+15i\sqrt{247} ki te 72.

$$x=\frac{175+5\sqrt{247}i}{24}$$

Nā, me whakaoti te whārite x=\frac{525±15\sqrt{247}i}{72} ina he tango te ±. Tango 15i\sqrt{247} mai i 525.

$$x=\frac{-15\sqrt{247}i+525}{72}$$

Whakawehe 525-15i\sqrt{247} ki te 72.

$$x=\frac{-5\sqrt{247}i+175}{24}$$

Kua oti te whārite te whakatau.

$$x=\frac{175+5\sqrt{247}i}{24} x=\frac{-5\sqrt{247}i+175}{24}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakareatia te 3 ki te 2, ka 6.

$$6\left(2x-10\right)\left(3x-30\right)=-5\left(3x+100\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 6 ki te 2x-10.

$$\left(12x-60\right)\left(3x-30\right)=-5\left(3x+100\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 12x-60 ki te 3x-30 ka whakakotahi i ngā kupu rite.

$$36x^{2}-540x+1800=-5\left(3x+100\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te -5 ki te 3x+100.

$$36x^{2}-540x+1800=-15x-500$$

Me tāpiri te 15x ki ngā taha e rua.

$$36x^{2}-540x+1800+15x=-500$$

Pahekotia te -540x me 15x, ka -525x.

$$36x^{2}-525x+1800=-500$$

Tangohia te 1800 mai i ngā taha e rua.

$$36x^{2}-525x=-500-1800$$

Tangohia te 1800 i te -500, ka -2300.

$$36x^{2}-525x=-2300$$

Whakawehea ngā taha e rua ki te 36.

$$\frac{36x^{2}-525x}{36}=-\frac{2300}{36}$$

Mā te whakawehe ki te 36 ka wetekia te whakareanga ki te 36.

$$x^{2}+\left(-\frac{525}{36}\right)x=-\frac{2300}{36}$$

Whakahekea te hautanga \frac{-525}{36} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 3.

$$x^{2}-\frac{175}{12}x=-\frac{2300}{36}$$

Whakahekea te hautanga \frac{-2300}{36} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

$$x^{2}-\frac{175}{12}x=-\frac{575}{9}$$

Whakawehea te -\frac{175}{12}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{175}{24}. Nā, tāpiria te pūrua o te -\frac{175}{24} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{175}{12}x+\left(-\frac{175}{24}\right)^{2}=-\frac{575}{9}+\left(-\frac{175}{24}\right)^{2}$$

Pūruatia -\frac{175}{24} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{175}{12}x+\frac{30625}{576}=-\frac{575}{9}+\frac{30625}{576}$$

Tāpiri -\frac{575}{9} ki te \frac{30625}{576} 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.

$$x^{2}-\frac{175}{12}x+\frac{30625}{576}=-\frac{6175}{576}$$

Tauwehea x^{2}-\frac{175}{12}x+\frac{30625}{576}. 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}.

$$\left(x-\frac{175}{24}\right)^{2}=-\frac{6175}{576}$$

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

$$\sqrt{\left(x-\frac{175}{24}\right)^{2}}=\sqrt{-\frac{6175}{576}}$$

Whakarūnātia.

$$x-\frac{175}{24}=\frac{5\sqrt{247}i}{24} x-\frac{175}{24}=-\frac{5\sqrt{247}i}{24}$$

Me tāpiri \frac{175}{24} ki ngā taha e rua o te whārite.

$$x=\frac{175+5\sqrt{247}i}{24} x=\frac{-5\sqrt{247}i+175}{24}$$