Whakaoti i te 2/x=4x+7/5x^2+1

$$\frac{ 2 }{ x } = \frac{ 4x+7 }{ 5 { x }^{ 2 } +1 }$$

Whakaoti mō x

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

$$x=\frac{2}{3}\approx 0.666666667$$

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. Me whakarea ngā taha e rua o te whārite ki te x\left(5x^{2}+1\right), arā, te tauraro pātahi he tino iti rawa te kitea o x,5x^{2}+1.

$$\left(5x^{2}+1\right)\times 2=x\left(4x+7\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 5x^{2}+1 ki te 2.

$$10x^{2}+2=x\left(4x+7\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x ki te 4x+7.

$$10x^{2}+2=4x^{2}+7x$$

Tangohia te 4x^{2} mai i ngā taha e rua.

$$10x^{2}+2-4x^{2}=7x$$

Pahekotia te 10x^{2} me -4x^{2}, ka 6x^{2}.

$$6x^{2}+2=7x$$

Tangohia te 7x mai i ngā taha e rua.

$$6x^{2}+2-7x=0$$

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

$$6x^{2}-7x+2=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 6x^{2}+ax+bx+2. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-7 ab=6\times 2=12$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōraro te a+b, he tōraro hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 12.

$$-1,-12 -2,-6 -3,-4$$

Tātaihia te tapeke mō ia takirua.

$$-1-12=-13 -2-6=-8 -3-4=-7$$

Ko te otinga te takirua ka hoatu i te tapeke -7.

$$a=-4 b=-3$$

Tuhia anō te 6x^{2}-7x+2 hei \left(6x^{2}-4x\right)+\left(-3x+2\right).

$$\left(6x^{2}-4x\right)+\left(-3x+2\right)$$

Tauwehea te 2x i te tuatahi me te -1 i te rōpū tuarua.

$$2x\left(3x-2\right)-\left(3x-2\right)$$

Whakatauwehea atu te kīanga pātahi 3x-2 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(3x-2\right)\left(2x-1\right)$$

Hei kimi otinga whārite, me whakaoti te 3x-2=0 me te 2x-1=0.

$$x=\frac{2}{3} x=\frac{1}{2}$$

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. Me whakarea ngā taha e rua o te whārite ki te x\left(5x^{2}+1\right), arā, te tauraro pātahi he tino iti rawa te kitea o x,5x^{2}+1.

$$\left(5x^{2}+1\right)\times 2=x\left(4x+7\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 5x^{2}+1 ki te 2.

$$10x^{2}+2=x\left(4x+7\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x ki te 4x+7.

$$10x^{2}+2=4x^{2}+7x$$

Tangohia te 4x^{2} mai i ngā taha e rua.

$$10x^{2}+2-4x^{2}=7x$$

Pahekotia te 10x^{2} me -4x^{2}, ka 6x^{2}.

$$6x^{2}+2=7x$$

Tangohia te 7x mai i ngā taha e rua.

$$6x^{2}+2-7x=0$$

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.

$$6x^{2}-7x+2=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 6 mō a, -7 mō b, me 2 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\times 6\times 2}}{2\times 6}$$

Pūrua -7.

$$x=\frac{-\left(-7\right)±\sqrt{49-4\times 6\times 2}}{2\times 6}$$

Whakareatia -4 ki te 6.

$$x=\frac{-\left(-7\right)±\sqrt{49-24\times 2}}{2\times 6}$$

Whakareatia -24 ki te 2.

$$x=\frac{-\left(-7\right)±\sqrt{49-48}}{2\times 6}$$

Tāpiri 49 ki te -48.

$$x=\frac{-\left(-7\right)±\sqrt{1}}{2\times 6}$$

Tuhia te pūtakerua o te 1.

$$x=\frac{-\left(-7\right)±1}{2\times 6}$$

Ko te tauaro o -7 ko 7.

$$x=\frac{7±1}{2\times 6}$$

Whakareatia 2 ki te 6.

$$x=\frac{7±1}{12}$$

Nā, me whakaoti te whārite x=\frac{7±1}{12} ina he tāpiri te ±. Tāpiri 7 ki te 1.

$$x=\frac{8}{12}$$

Whakahekea te hautanga \frac{8}{12} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

$$x=\frac{2}{3}$$

Nā, me whakaoti te whārite x=\frac{7±1}{12} ina he tango te ±. Tango 1 mai i 7.

$$x=\frac{6}{12}$$

Whakahekea te hautanga \frac{6}{12} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 6.

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

Kua oti te whārite te whakatau.

$$x=\frac{2}{3} x=\frac{1}{2}$$

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. Me whakarea ngā taha e rua o te whārite ki te x\left(5x^{2}+1\right), arā, te tauraro pātahi he tino iti rawa te kitea o x,5x^{2}+1.

$$\left(5x^{2}+1\right)\times 2=x\left(4x+7\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 5x^{2}+1 ki te 2.

$$10x^{2}+2=x\left(4x+7\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x ki te 4x+7.

$$10x^{2}+2=4x^{2}+7x$$

Tangohia te 4x^{2} mai i ngā taha e rua.

$$10x^{2}+2-4x^{2}=7x$$

Pahekotia te 10x^{2} me -4x^{2}, ka 6x^{2}.

$$6x^{2}+2=7x$$

Tangohia te 7x mai i ngā taha e rua.

$$6x^{2}+2-7x=0$$

Tangohia te 2 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$6x^{2}-7x=-2$$

Whakawehea ngā taha e rua ki te 6.

$$\frac{6x^{2}-7x}{6}=-\frac{2}{6}$$

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

$$x^{2}-\frac{7}{6}x=-\frac{2}{6}$$

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

$$x^{2}-\frac{7}{6}x=-\frac{1}{3}$$

Whakawehea te -\frac{7}{6}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{7}{12}. Nā, tāpiria te pūrua o te -\frac{7}{12} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{7}{6}x+\left(-\frac{7}{12}\right)^{2}=-\frac{1}{3}+\left(-\frac{7}{12}\right)^{2}$$

Pūruatia -\frac{7}{12} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{7}{6}x+\frac{49}{144}=-\frac{1}{3}+\frac{49}{144}$$

Tāpiri -\frac{1}{3} ki te \frac{49}{144} 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.

$$x^{2}-\frac{7}{6}x+\frac{49}{144}=\frac{1}{144}$$

Tauwehea te x^{2}-\frac{7}{6}x+\frac{49}{144}. 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}.

$$\left(x-\frac{7}{12}\right)^{2}=\frac{1}{144}$$

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

$$\sqrt{\left(x-\frac{7}{12}\right)^{2}}=\sqrt{\frac{1}{144}}$$

Whakarūnātia.

$$x-\frac{7}{12}=\frac{1}{12} x-\frac{7}{12}=-\frac{1}{12}$$

Me tāpiri \frac{7}{12} ki ngā taha e rua o te whārite.

$$x=\frac{2}{3} x=\frac{1}{2}$$