Whakaoti i te (2x-1)(-3x+4)=-6x+11x+4

$$( 2 x - 1 ) ( - 3 x + 4 ) = - 6 x + 11 x + 4$$

Whakaoti mō x (complex solution)

$$x=\frac{\sqrt{39}i}{6}+\frac{1}{2}\approx 0.5+1.040833i$$

$$x=-\frac{\sqrt{39}i}{6}+\frac{1}{2}\approx 0.5-1.040833i$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te 2x-1 ki te -3x+4 ka whakakotahi i ngā kupu rite.

$$-6x^{2}+11x-4=-6x+11x+4$$

Pahekotia te -6x me 11x, ka 5x.

$$-6x^{2}+11x-4=5x+4$$

Tangohia te 5x mai i ngā taha e rua.

$$-6x^{2}+11x-4-5x=4$$

Pahekotia te 11x me -5x, ka 6x.

$$-6x^{2}+6x-4=4$$

Tangohia te 4 mai i ngā taha e rua.

$$-6x^{2}+6x-4-4=0$$

Tangohia te 4 i te -4, ka -8.

$$-6x^{2}+6x-8=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -6 mō a, 6 mō b, me -8 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-6±\sqrt{6^{2}-4\left(-6\right)\left(-8\right)}}{2\left(-6\right)}$$

Pūrua 6.

$$x=\frac{-6±\sqrt{36-4\left(-6\right)\left(-8\right)}}{2\left(-6\right)}$$

Whakareatia -4 ki te -6.

$$x=\frac{-6±\sqrt{36+24\left(-8\right)}}{2\left(-6\right)}$$

Whakareatia 24 ki te -8.

$$x=\frac{-6±\sqrt{36-192}}{2\left(-6\right)}$$

Tāpiri 36 ki te -192.

$$x=\frac{-6±\sqrt{-156}}{2\left(-6\right)}$$

Tuhia te pūtakerua o te -156.

$$x=\frac{-6±2\sqrt{39}i}{2\left(-6\right)}$$

Whakareatia 2 ki te -6.

$$x=\frac{-6±2\sqrt{39}i}{-12}$$

Nā, me whakaoti te whārite x=\frac{-6±2\sqrt{39}i}{-12} ina he tāpiri te ±. Tāpiri -6 ki te 2i\sqrt{39}.

$$x=\frac{-6+2\sqrt{39}i}{-12}$$

Whakawehe -6+2i\sqrt{39} ki te -12.

$$x=-\frac{\sqrt{39}i}{6}+\frac{1}{2}$$

Nā, me whakaoti te whārite x=\frac{-6±2\sqrt{39}i}{-12} ina he tango te ±. Tango 2i\sqrt{39} mai i -6.

$$x=\frac{-2\sqrt{39}i-6}{-12}$$

Whakawehe -6-2i\sqrt{39} ki te -12.

$$x=\frac{\sqrt{39}i}{6}+\frac{1}{2}$$

Kua oti te whārite te whakatau.

$$x=-\frac{\sqrt{39}i}{6}+\frac{1}{2} x=\frac{\sqrt{39}i}{6}+\frac{1}{2}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te 2x-1 ki te -3x+4 ka whakakotahi i ngā kupu rite.

$$-6x^{2}+11x-4=-6x+11x+4$$

Pahekotia te -6x me 11x, ka 5x.

$$-6x^{2}+11x-4=5x+4$$

Tangohia te 5x mai i ngā taha e rua.

$$-6x^{2}+11x-4-5x=4$$

Pahekotia te 11x me -5x, ka 6x.

$$-6x^{2}+6x-4=4$$

Me tāpiri te 4 ki ngā taha e rua.

$$-6x^{2}+6x=4+4$$

Tāpirihia te 4 ki te 4, ka 8.

$$-6x^{2}+6x=8$$

Whakawehea ngā taha e rua ki te -6.

$$\frac{-6x^{2}+6x}{-6}=\frac{8}{-6}$$

Mā te whakawehe ki te -6 ka wetekia te whakareanga ki te -6.

$$x^{2}+\frac{6}{-6}x=\frac{8}{-6}$$

Whakawehe 6 ki te -6.

$$x^{2}-x=\frac{8}{-6}$$

Whakahekea te hautanga \frac{8}{-6} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x^{2}-x=-\frac{4}{3}$$

Whakawehea te -1, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{1}{2}. Nā, tāpiria te pūrua o te -\frac{1}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-x+\left(-\frac{1}{2}\right)^{2}=-\frac{4}{3}+\left(-\frac{1}{2}\right)^{2}$$

Pūruatia -\frac{1}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-x+\frac{1}{4}=-\frac{4}{3}+\frac{1}{4}$$

Tāpiri -\frac{4}{3} ki te \frac{1}{4} 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.

$$x^{2}-x+\frac{1}{4}=-\frac{13}{12}$$

Tauwehea x^{2}-x+\frac{1}{4}. 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}.

$$\left(x-\frac{1}{2}\right)^{2}=-\frac{13}{12}$$

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

$$\sqrt{\left(x-\frac{1}{2}\right)^{2}}=\sqrt{-\frac{13}{12}}$$

Whakarūnātia.

$$x-\frac{1}{2}=\frac{\sqrt{39}i}{6} x-\frac{1}{2}=-\frac{\sqrt{39}i}{6}$$

Me tāpiri \frac{1}{2} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{39}i}{6}+\frac{1}{2} x=-\frac{\sqrt{39}i}{6}+\frac{1}{2}$$