Whakaoti i te 14-(5x-1)(2x+3)=17-(10x+19(x-6)

$$14 - ( 5 x - 1 ) ( 2 x + 3 ) = 17 - ( 10 x + 19 ( x - 6 )$$

Whakaoti mō x (complex solution)

$$x=\frac{4+\sqrt{269}i}{5}\approx 0.8+3.280243893i$$

$$x=\frac{-\sqrt{269}i+4}{5}\approx 0.8-3.280243893i$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te 5x-1 ki te 2x+3 ka whakakotahi i ngā kupu rite.

$$14-\left(10x^{2}+13x-3\right)=17-\left(10x+19\left(x-6\right)\right)$$

Hei kimi i te tauaro o 10x^{2}+13x-3, kimihia te tauaro o ia taurangi.

$$14-10x^{2}-13x+3=17-\left(10x+19\left(x-6\right)\right)$$

Tāpirihia te 14 ki te 3, ka 17.

$$17-10x^{2}-13x=17-\left(10x+19\left(x-6\right)\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 19 ki te x-6.

$$17-10x^{2}-13x=17-\left(10x+19x-114\right)$$

Pahekotia te 10x me 19x, ka 29x.

$$17-10x^{2}-13x=17-\left(29x-114\right)$$

Hei kimi i te tauaro o 29x-114, kimihia te tauaro o ia taurangi.

$$17-10x^{2}-13x=17-29x+114$$

Tāpirihia te 17 ki te 114, ka 131.

$$17-10x^{2}-13x=131-29x$$

Tangohia te 131 mai i ngā taha e rua.

$$17-10x^{2}-13x-131=-29x$$

Tangohia te 131 i te 17, ka -114.

$$-114-10x^{2}-13x=-29x$$

Me tāpiri te 29x ki ngā taha e rua.

$$-114-10x^{2}-13x+29x=0$$

Pahekotia te -13x me 29x, ka 16x.

$$-114-10x^{2}+16x=0$$

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.

$$-10x^{2}+16x-114=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -10 mō a, 16 mō b, me -114 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-16±\sqrt{16^{2}-4\left(-10\right)\left(-114\right)}}{2\left(-10\right)}$$

Pūrua 16.

$$x=\frac{-16±\sqrt{256-4\left(-10\right)\left(-114\right)}}{2\left(-10\right)}$$

Whakareatia -4 ki te -10.

$$x=\frac{-16±\sqrt{256+40\left(-114\right)}}{2\left(-10\right)}$$

Whakareatia 40 ki te -114.

$$x=\frac{-16±\sqrt{256-4560}}{2\left(-10\right)}$$

Tāpiri 256 ki te -4560.

$$x=\frac{-16±\sqrt{-4304}}{2\left(-10\right)}$$

Tuhia te pūtakerua o te -4304.

$$x=\frac{-16±4\sqrt{269}i}{2\left(-10\right)}$$

Whakareatia 2 ki te -10.

$$x=\frac{-16±4\sqrt{269}i}{-20}$$

Nā, me whakaoti te whārite x=\frac{-16±4\sqrt{269}i}{-20} ina he tāpiri te ±. Tāpiri -16 ki te 4i\sqrt{269}.

$$x=\frac{-16+4\sqrt{269}i}{-20}$$

Whakawehe -16+4i\sqrt{269} ki te -20.

$$x=\frac{-\sqrt{269}i+4}{5}$$

Nā, me whakaoti te whārite x=\frac{-16±4\sqrt{269}i}{-20} ina he tango te ±. Tango 4i\sqrt{269} mai i -16.

$$x=\frac{-4\sqrt{269}i-16}{-20}$$

Whakawehe -16-4i\sqrt{269} ki te -20.

$$x=\frac{4+\sqrt{269}i}{5}$$

Kua oti te whārite te whakatau.

$$x=\frac{-\sqrt{269}i+4}{5} x=\frac{4+\sqrt{269}i}{5}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te 5x-1 ki te 2x+3 ka whakakotahi i ngā kupu rite.

$$14-\left(10x^{2}+13x-3\right)=17-\left(10x+19\left(x-6\right)\right)$$

Hei kimi i te tauaro o 10x^{2}+13x-3, kimihia te tauaro o ia taurangi.

$$14-10x^{2}-13x+3=17-\left(10x+19\left(x-6\right)\right)$$

Tāpirihia te 14 ki te 3, ka 17.

$$17-10x^{2}-13x=17-\left(10x+19\left(x-6\right)\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 19 ki te x-6.

$$17-10x^{2}-13x=17-\left(10x+19x-114\right)$$

Pahekotia te 10x me 19x, ka 29x.

$$17-10x^{2}-13x=17-\left(29x-114\right)$$

Hei kimi i te tauaro o 29x-114, kimihia te tauaro o ia taurangi.

$$17-10x^{2}-13x=17-29x+114$$

Tāpirihia te 17 ki te 114, ka 131.

$$17-10x^{2}-13x=131-29x$$

Me tāpiri te 29x ki ngā taha e rua.

$$17-10x^{2}-13x+29x=131$$

Pahekotia te -13x me 29x, ka 16x.

$$17-10x^{2}+16x=131$$

Tangohia te 17 mai i ngā taha e rua.

$$-10x^{2}+16x=131-17$$

Tangohia te 17 i te 131, ka 114.

$$-10x^{2}+16x=114$$

Whakawehea ngā taha e rua ki te -10.

$$\frac{-10x^{2}+16x}{-10}=\frac{114}{-10}$$

Mā te whakawehe ki te -10 ka wetekia te whakareanga ki te -10.

$$x^{2}+\frac{16}{-10}x=\frac{114}{-10}$$

Whakahekea te hautanga \frac{16}{-10} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x^{2}-\frac{8}{5}x=\frac{114}{-10}$$

Whakahekea te hautanga \frac{114}{-10} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x^{2}-\frac{8}{5}x=-\frac{57}{5}$$

Whakawehea te -\frac{8}{5}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{4}{5}. Nā, tāpiria te pūrua o te -\frac{4}{5} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{8}{5}x+\left(-\frac{4}{5}\right)^{2}=-\frac{57}{5}+\left(-\frac{4}{5}\right)^{2}$$

Pūruatia -\frac{4}{5} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{8}{5}x+\frac{16}{25}=-\frac{57}{5}+\frac{16}{25}$$

Tāpiri -\frac{57}{5} ki te \frac{16}{25} 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.

$$x^{2}-\frac{8}{5}x+\frac{16}{25}=-\frac{269}{25}$$

Tauwehea x^{2}-\frac{8}{5}x+\frac{16}{25}. 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}.

$$\left(x-\frac{4}{5}\right)^{2}=-\frac{269}{25}$$

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

$$\sqrt{\left(x-\frac{4}{5}\right)^{2}}=\sqrt{-\frac{269}{25}}$$

Whakarūnātia.

$$x-\frac{4}{5}=\frac{\sqrt{269}i}{5} x-\frac{4}{5}=-\frac{\sqrt{269}i}{5}$$

Me tāpiri \frac{4}{5} ki ngā taha e rua o te whārite.

$$x=\frac{4+\sqrt{269}i}{5} x=\frac{-\sqrt{269}i+4}{5}$$