Whakaoti i te 4x*2x-x=12*7-2

$$4x \times 2x-x=12 \times 7-2$$

Whakaoti mō x

$$x = \frac{5 \sqrt{105} + 1}{16} \approx 3.264672114$$

$$x=\frac{1-5\sqrt{105}}{16}\approx -3.139672114$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakareatia te x ki te x, ka x^{2}.

$$4x^{2}\times 2-x=12\times 7-2$$

Whakareatia te 4 ki te 2, ka 8.

$$8x^{2}-x=12\times 7-2$$

Whakareatia te 12 ki te 7, ka 84.

$$8x^{2}-x=84-2$$

Tangohia te 2 i te 84, ka 82.

$$8x^{2}-x=82$$

Tangohia te 82 mai i ngā taha e rua.

$$8x^{2}-x-82=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 8 mō a, -1 mō b, me -82 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\times 8\left(-82\right)}}{2\times 8}$$

Whakareatia -4 ki te 8.

$$x=\frac{-\left(-1\right)±\sqrt{1-32\left(-82\right)}}{2\times 8}$$

Whakareatia -32 ki te -82.

$$x=\frac{-\left(-1\right)±\sqrt{1+2624}}{2\times 8}$$

Tāpiri 1 ki te 2624.

$$x=\frac{-\left(-1\right)±\sqrt{2625}}{2\times 8}$$

Tuhia te pūtakerua o te 2625.

$$x=\frac{-\left(-1\right)±5\sqrt{105}}{2\times 8}$$

Ko te tauaro o -1 ko 1.

$$x=\frac{1±5\sqrt{105}}{2\times 8}$$

Whakareatia 2 ki te 8.

$$x=\frac{1±5\sqrt{105}}{16}$$

Nā, me whakaoti te whārite x=\frac{1±5\sqrt{105}}{16} ina he tāpiri te ±. Tāpiri 1 ki te 5\sqrt{105}.

$$x=\frac{5\sqrt{105}+1}{16}$$

Nā, me whakaoti te whārite x=\frac{1±5\sqrt{105}}{16} ina he tango te ±. Tango 5\sqrt{105} mai i 1.

$$x=\frac{1-5\sqrt{105}}{16}$$

Kua oti te whārite te whakatau.

$$x=\frac{5\sqrt{105}+1}{16} x=\frac{1-5\sqrt{105}}{16}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakareatia te x ki te x, ka x^{2}.

$$4x^{2}\times 2-x=12\times 7-2$$

Whakareatia te 4 ki te 2, ka 8.

$$8x^{2}-x=12\times 7-2$$

Whakareatia te 12 ki te 7, ka 84.

$$8x^{2}-x=84-2$$

Tangohia te 2 i te 84, ka 82.

$$8x^{2}-x=82$$

Whakawehea ngā taha e rua ki te 8.

$$\frac{8x^{2}-x}{8}=\frac{82}{8}$$

Mā te whakawehe ki te 8 ka wetekia te whakareanga ki te 8.

$$x^{2}-\frac{1}{8}x=\frac{82}{8}$$

Whakahekea te hautanga \frac{82}{8} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x^{2}-\frac{1}{8}x=\frac{41}{4}$$

Whakawehea te -\frac{1}{8}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{1}{16}. Nā, tāpiria te pūrua o te -\frac{1}{16} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{1}{8}x+\left(-\frac{1}{16}\right)^{2}=\frac{41}{4}+\left(-\frac{1}{16}\right)^{2}$$

Pūruatia -\frac{1}{16} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{1}{8}x+\frac{1}{256}=\frac{41}{4}+\frac{1}{256}$$

Tāpiri \frac{41}{4} ki te \frac{1}{256} 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.

$$x^{2}-\frac{1}{8}x+\frac{1}{256}=\frac{2625}{256}$$

Tauwehea x^{2}-\frac{1}{8}x+\frac{1}{256}. 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}.

$$\left(x-\frac{1}{16}\right)^{2}=\frac{2625}{256}$$

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

$$\sqrt{\left(x-\frac{1}{16}\right)^{2}}=\sqrt{\frac{2625}{256}}$$

Whakarūnātia.

$$x-\frac{1}{16}=\frac{5\sqrt{105}}{16} x-\frac{1}{16}=-\frac{5\sqrt{105}}{16}$$

Me tāpiri \frac{1}{16} ki ngā taha e rua o te whārite.

$$x=\frac{5\sqrt{105}+1}{16} x=\frac{1-5\sqrt{105}}{16}$$