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

$$( 4 x - 1 ) ^ { 2 } = ( x - 1 ) ( x + 1 )$$

Whakaoti mō x (complex solution)

$$x=\frac{4+\sqrt{14}i}{15}\approx 0.266666667+0.249443826i$$

$$x=\frac{-\sqrt{14}i+4}{15}\approx 0.266666667-0.249443826i$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(4x-1\right)^{2}.

$$16x^{2}-8x+1=\left(x-1\right)\left(x+1\right)$$

Whakaarohia te \left(x-1\right)\left(x+1\right). Ka taea te whakareanga te panoni ki te rerekētanga o ngā pūrua mā te ture: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Pūrua 1.

$$16x^{2}-8x+1=x^{2}-1$$

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

$$16x^{2}-8x+1-x^{2}=-1$$

Pahekotia te 16x^{2} me -x^{2}, ka 15x^{2}.

$$15x^{2}-8x+1=-1$$

Me tāpiri te 1 ki ngā taha e rua.

$$15x^{2}-8x+1+1=0$$

Tāpirihia te 1 ki te 1, ka 2.

$$15x^{2}-8x+2=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 15 mō a, -8 mō b, me 2 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\times 15\times 2}}{2\times 15}$$

Pūrua -8.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\times 15\times 2}}{2\times 15}$$

Whakareatia -4 ki te 15.

$$x=\frac{-\left(-8\right)±\sqrt{64-60\times 2}}{2\times 15}$$

Whakareatia -60 ki te 2.

$$x=\frac{-\left(-8\right)±\sqrt{64-120}}{2\times 15}$$

Tāpiri 64 ki te -120.

$$x=\frac{-\left(-8\right)±\sqrt{-56}}{2\times 15}$$

Tuhia te pūtakerua o te -56.

$$x=\frac{-\left(-8\right)±2\sqrt{14}i}{2\times 15}$$

Ko te tauaro o -8 ko 8.

$$x=\frac{8±2\sqrt{14}i}{2\times 15}$$

Whakareatia 2 ki te 15.

$$x=\frac{8±2\sqrt{14}i}{30}$$

Nā, me whakaoti te whārite x=\frac{8±2\sqrt{14}i}{30} ina he tāpiri te ±. Tāpiri 8 ki te 2i\sqrt{14}.

$$x=\frac{8+2\sqrt{14}i}{30}$$

Whakawehe 8+2i\sqrt{14} ki te 30.

$$x=\frac{4+\sqrt{14}i}{15}$$

Nā, me whakaoti te whārite x=\frac{8±2\sqrt{14}i}{30} ina he tango te ±. Tango 2i\sqrt{14} mai i 8.

$$x=\frac{-2\sqrt{14}i+8}{30}$$

Whakawehe 8-2i\sqrt{14} ki te 30.

$$x=\frac{-\sqrt{14}i+4}{15}$$

Kua oti te whārite te whakatau.

$$x=\frac{4+\sqrt{14}i}{15} x=\frac{-\sqrt{14}i+4}{15}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(4x-1\right)^{2}.

$$16x^{2}-8x+1=\left(x-1\right)\left(x+1\right)$$

Whakaarohia te \left(x-1\right)\left(x+1\right). Ka taea te whakareanga te panoni ki te rerekētanga o ngā pūrua mā te ture: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Pūrua 1.

$$16x^{2}-8x+1=x^{2}-1$$

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

$$16x^{2}-8x+1-x^{2}=-1$$

Pahekotia te 16x^{2} me -x^{2}, ka 15x^{2}.

$$15x^{2}-8x+1=-1$$

Tangohia te 1 mai i ngā taha e rua.

$$15x^{2}-8x=-1-1$$

Tangohia te 1 i te -1, ka -2.

$$15x^{2}-8x=-2$$

Whakawehea ngā taha e rua ki te 15.

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

Mā te whakawehe ki te 15 ka wetekia te whakareanga ki te 15.

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

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.

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

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

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

Tāpiri -\frac{2}{15} 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.

$$x^{2}-\frac{8}{15}x+\frac{16}{225}=-\frac{14}{225}$$

Tauwehea te x^{2}-\frac{8}{15}x+\frac{16}{225}. Ko te tikanga, ina ko x^{2}+bx+c he pūrua tika, ka taea te tauwehe i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{4}{15}\right)^{2}=-\frac{14}{225}$$

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

$$\sqrt{\left(x-\frac{4}{15}\right)^{2}}=\sqrt{-\frac{14}{225}}$$

Whakarūnātia.

$$x-\frac{4}{15}=\frac{\sqrt{14}i}{15} x-\frac{4}{15}=-\frac{\sqrt{14}i}{15}$$

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

$$x=\frac{4+\sqrt{14}i}{15} x=\frac{-\sqrt{14}i+4}{15}$$