Whakaoti i te 34x^2-24x-1/(x+1)(x-1)=0

$$\frac { 34 x ^ { 2 } - 24 x - 1 } { ( x + 1 ) ( x - 1 ) } = 0$$

Whakaoti mō x

$$x=\frac{\sqrt{178}}{34}+\frac{6}{17}\approx 0.745343061$$

$$x=-\frac{\sqrt{178}}{34}+\frac{6}{17}\approx -0.039460708$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -1,1 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te \left(x-1\right)\left(x+1\right).

$$34x^{2}-24x-1=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 34 mō a, -24 mō b, me -1 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-24\right)±\sqrt{\left(-24\right)^{2}-4\times 34\left(-1\right)}}{2\times 34}$$

Pūrua -24.

$$x=\frac{-\left(-24\right)±\sqrt{576-4\times 34\left(-1\right)}}{2\times 34}$$

Whakareatia -4 ki te 34.

$$x=\frac{-\left(-24\right)±\sqrt{576-136\left(-1\right)}}{2\times 34}$$

Whakareatia -136 ki te -1.

$$x=\frac{-\left(-24\right)±\sqrt{576+136}}{2\times 34}$$

Tāpiri 576 ki te 136.

$$x=\frac{-\left(-24\right)±\sqrt{712}}{2\times 34}$$

Tuhia te pūtakerua o te 712.

$$x=\frac{-\left(-24\right)±2\sqrt{178}}{2\times 34}$$

Ko te tauaro o -24 ko 24.

$$x=\frac{24±2\sqrt{178}}{2\times 34}$$

Whakareatia 2 ki te 34.

$$x=\frac{24±2\sqrt{178}}{68}$$

Nā, me whakaoti te whārite x=\frac{24±2\sqrt{178}}{68} ina he tāpiri te ±. Tāpiri 24 ki te 2\sqrt{178}.

$$x=\frac{2\sqrt{178}+24}{68}$$

Whakawehe 24+2\sqrt{178} ki te 68.

$$x=\frac{\sqrt{178}}{34}+\frac{6}{17}$$

Nā, me whakaoti te whārite x=\frac{24±2\sqrt{178}}{68} ina he tango te ±. Tango 2\sqrt{178} mai i 24.

$$x=\frac{24-2\sqrt{178}}{68}$$

Whakawehe 24-2\sqrt{178} ki te 68.

$$x=-\frac{\sqrt{178}}{34}+\frac{6}{17}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{178}}{34}+\frac{6}{17} x=-\frac{\sqrt{178}}{34}+\frac{6}{17}$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -1,1 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te \left(x-1\right)\left(x+1\right).

$$34x^{2}-24x-1=0$$

Me tāpiri te 1 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$34x^{2}-24x=1$$

Whakawehea ngā taha e rua ki te 34.

$$\frac{34x^{2}-24x}{34}=\frac{1}{34}$$

Mā te whakawehe ki te 34 ka wetekia te whakareanga ki te 34.

$$x^{2}+\left(-\frac{24}{34}\right)x=\frac{1}{34}$$

Whakahekea te hautanga \frac{-24}{34} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x^{2}-\frac{12}{17}x=\frac{1}{34}$$

Whakawehea te -\frac{12}{17}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{6}{17}. Nā, tāpiria te pūrua o te -\frac{6}{17} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{12}{17}x+\left(-\frac{6}{17}\right)^{2}=\frac{1}{34}+\left(-\frac{6}{17}\right)^{2}$$

Pūruatia -\frac{6}{17} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{12}{17}x+\frac{36}{289}=\frac{1}{34}+\frac{36}{289}$$

Tāpiri \frac{1}{34} ki te \frac{36}{289} 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.

$$x^{2}-\frac{12}{17}x+\frac{36}{289}=\frac{89}{578}$$

Tauwehea x^{2}-\frac{12}{17}x+\frac{36}{289}. 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}.

$$\left(x-\frac{6}{17}\right)^{2}=\frac{89}{578}$$

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

$$\sqrt{\left(x-\frac{6}{17}\right)^{2}}=\sqrt{\frac{89}{578}}$$

Whakarūnātia.

$$x-\frac{6}{17}=\frac{\sqrt{178}}{34} x-\frac{6}{17}=-\frac{\sqrt{178}}{34}$$

Me tāpiri \frac{6}{17} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{178}}{34}+\frac{6}{17} x=-\frac{\sqrt{178}}{34}+\frac{6}{17}$$