Whakaoti i te 10x-1/x=3.xneq0

$$10 x - \frac { 1 } { x } = 3 . x \neq 0$$

Whakaoti mō x

$$x=-\frac{1}{5}=-0.2$$

$$x=\frac{1}{2}=0.5$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tē taea kia ōrite te tāupe x ki 0 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te x.

$$10xx-1=3x$$

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

$$10x^{2}-1=3x$$

Tangohia te 3x mai i ngā taha e rua.

$$10x^{2}-1-3x=0$$

Hurinahatia te pūrau ki te āhua tānga ngahuru. Whakaraupapahia ngā kīanga tau mai i te pū teitei rawa ki te mea iti rawa.

$$10x^{2}-3x-1=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 10x^{2}+ax+bx-1. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-3 ab=10\left(-1\right)=-10$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōraro te a+b, he nui ake te uara pū o te tau tōraro i tō te tōrunga. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -10.

$$1,-10 2,-5$$

Tātaihia te tapeke mō ia takirua.

$$1-10=-9 2-5=-3$$

Ko te otinga te takirua ka hoatu i te tapeke -3.

$$a=-5 b=2$$

Tuhia anō te 10x^{2}-3x-1 hei \left(10x^{2}-5x\right)+\left(2x-1\right).

$$\left(10x^{2}-5x\right)+\left(2x-1\right)$$

Whakatauwehea atu 5x i te 10x^{2}-5x.

$$5x\left(2x-1\right)+2x-1$$

Whakatauwehea atu te kīanga pātahi 2x-1 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(2x-1\right)\left(5x+1\right)$$

Hei kimi otinga whārite, me whakaoti te 2x-1=0 me te 5x+1=0.

$$x=\frac{1}{2} x=-\frac{1}{5}$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki 0 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te x.

$$10xx-1=3x$$

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

$$10x^{2}-1=3x$$

Tangohia te 3x mai i ngā taha e rua.

$$10x^{2}-1-3x=0$$

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.

$$10x^{2}-3x-1=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 10 mō a, -3 mō b, me -1 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\times 10\left(-1\right)}}{2\times 10}$$

Pūrua -3.

$$x=\frac{-\left(-3\right)±\sqrt{9-4\times 10\left(-1\right)}}{2\times 10}$$

Whakareatia -4 ki te 10.

$$x=\frac{-\left(-3\right)±\sqrt{9-40\left(-1\right)}}{2\times 10}$$

Whakareatia -40 ki te -1.

$$x=\frac{-\left(-3\right)±\sqrt{9+40}}{2\times 10}$$

Tāpiri 9 ki te 40.

$$x=\frac{-\left(-3\right)±\sqrt{49}}{2\times 10}$$

Tuhia te pūtakerua o te 49.

$$x=\frac{-\left(-3\right)±7}{2\times 10}$$

Ko te tauaro o -3 ko 3.

$$x=\frac{3±7}{2\times 10}$$

Whakareatia 2 ki te 10.

$$x=\frac{3±7}{20}$$

Nā, me whakaoti te whārite x=\frac{3±7}{20} ina he tāpiri te ±. Tāpiri 3 ki te 7.

$$x=\frac{10}{20}$$

Whakahekea te hautanga \frac{10}{20} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 10.

$$x=\frac{1}{2}$$

Nā, me whakaoti te whārite x=\frac{3±7}{20} ina he tango te ±. Tango 7 mai i 3.

$$x=-\frac{4}{20}$$

Whakahekea te hautanga \frac{-4}{20} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

$$x=-\frac{1}{5}$$

Kua oti te whārite te whakatau.

$$x=\frac{1}{2} x=-\frac{1}{5}$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki 0 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te x.

$$10xx-1=3x$$

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

$$10x^{2}-1=3x$$

Tangohia te 3x mai i ngā taha e rua.

$$10x^{2}-1-3x=0$$

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

$$10x^{2}-3x=1$$

Whakawehea ngā taha e rua ki te 10.

$$\frac{10x^{2}-3x}{10}=\frac{1}{10}$$

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

$$x^{2}-\frac{3}{10}x=\frac{1}{10}$$

Whakawehea te -\frac{3}{10}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{3}{20}. Nā, tāpiria te pūrua o te -\frac{3}{20} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{3}{10}x+\left(-\frac{3}{20}\right)^{2}=\frac{1}{10}+\left(-\frac{3}{20}\right)^{2}$$

Pūruatia -\frac{3}{20} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{3}{10}x+\frac{9}{400}=\frac{1}{10}+\frac{9}{400}$$

Tāpiri \frac{1}{10} ki te \frac{9}{400} 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.

$$x^{2}-\frac{3}{10}x+\frac{9}{400}=\frac{49}{400}$$

Tauwehea x^{2}-\frac{3}{10}x+\frac{9}{400}. 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}.

$$\left(x-\frac{3}{20}\right)^{2}=\frac{49}{400}$$

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

$$\sqrt{\left(x-\frac{3}{20}\right)^{2}}=\sqrt{\frac{49}{400}}$$

Whakarūnātia.

$$x-\frac{3}{20}=\frac{7}{20} x-\frac{3}{20}=-\frac{7}{20}$$

Me tāpiri \frac{3}{20} ki ngā taha e rua o te whārite.

$$x=\frac{1}{2} x=-\frac{1}{5}$$