Whakaoti i te 4-8/3x+1=3x^2+5/3x+1

$$4 - \frac { 8 } { 3 x + 1 } = \frac { 3 x ^ { 2 } + 5 } { 3 x + 1 }$$

Whakaoti mō x

$$x=1$$

$$x=3$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tē taea kia ōrite te tāupe x ki -\frac{1}{3} nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te 3x+1.

$$\left(3x+1\right)\times 4-8=3x^{2}+5$$

Whakamahia te āhuatanga tohatoha hei whakarea te 3x+1 ki te 4.

$$12x+4-8=3x^{2}+5$$

Tangohia te 8 i te 4, ka -4.

$$12x-4=3x^{2}+5$$

Tangohia te 3x^{2} mai i ngā taha e rua.

$$12x-4-3x^{2}=5$$

Tangohia te 5 mai i ngā taha e rua.

$$12x-4-3x^{2}-5=0$$

Tangohia te 5 i te -4, ka -9.

$$12x-9-3x^{2}=0$$

Whakawehea ngā taha e rua ki te 3.

$$4x-3-x^{2}=0$$

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

$$-x^{2}+4x-3=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei -x^{2}+ax+bx-3. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=4 ab=-\left(-3\right)=3$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōrunga te a+b, he tōrunga hoki a a me b. Ko te takirua anake pērā ko te otinga pūnaha.

$$a=3 b=1$$

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

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

Whakatauwehea atu -x i te -x^{2}+3x.

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

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

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

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

$$x=3 x=1$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki -\frac{1}{3} nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te 3x+1.

$$\left(3x+1\right)\times 4-8=3x^{2}+5$$

Whakamahia te āhuatanga tohatoha hei whakarea te 3x+1 ki te 4.

$$12x+4-8=3x^{2}+5$$

Tangohia te 8 i te 4, ka -4.

$$12x-4=3x^{2}+5$$

Tangohia te 3x^{2} mai i ngā taha e rua.

$$12x-4-3x^{2}=5$$

Tangohia te 5 mai i ngā taha e rua.

$$12x-4-3x^{2}-5=0$$

Tangohia te 5 i te -4, ka -9.

$$12x-9-3x^{2}=0$$

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.

$$-3x^{2}+12x-9=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -3 mō a, 12 mō b, me -9 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-12±\sqrt{12^{2}-4\left(-3\right)\left(-9\right)}}{2\left(-3\right)}$$

Pūrua 12.

$$x=\frac{-12±\sqrt{144-4\left(-3\right)\left(-9\right)}}{2\left(-3\right)}$$

Whakareatia -4 ki te -3.

$$x=\frac{-12±\sqrt{144+12\left(-9\right)}}{2\left(-3\right)}$$

Whakareatia 12 ki te -9.

$$x=\frac{-12±\sqrt{144-108}}{2\left(-3\right)}$$

Tāpiri 144 ki te -108.

$$x=\frac{-12±\sqrt{36}}{2\left(-3\right)}$$

Tuhia te pūtakerua o te 36.

$$x=\frac{-12±6}{2\left(-3\right)}$$

Whakareatia 2 ki te -3.

$$x=\frac{-12±6}{-6}$$

Nā, me whakaoti te whārite x=\frac{-12±6}{-6} ina he tāpiri te ±. Tāpiri -12 ki te 6.

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

Whakawehe -6 ki te -6.

$$x=1$$

Nā, me whakaoti te whārite x=\frac{-12±6}{-6} ina he tango te ±. Tango 6 mai i -12.

$$x=-\frac{18}{-6}$$

Whakawehe -18 ki te -6.

$$x=3$$

Kua oti te whārite te whakatau.

$$x=1 x=3$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki -\frac{1}{3} nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te 3x+1.

$$\left(3x+1\right)\times 4-8=3x^{2}+5$$

Whakamahia te āhuatanga tohatoha hei whakarea te 3x+1 ki te 4.

$$12x+4-8=3x^{2}+5$$

Tangohia te 8 i te 4, ka -4.

$$12x-4=3x^{2}+5$$

Tangohia te 3x^{2} mai i ngā taha e rua.

$$12x-4-3x^{2}=5$$

Me tāpiri te 4 ki ngā taha e rua.

$$12x-3x^{2}=5+4$$

Tāpirihia te 5 ki te 4, ka 9.

$$12x-3x^{2}=9$$

Ko ngā whārite pūrua pēnei i tēnei nā ka taea te whakaoti mā te whakaoti i te pūrua. Hei whakaoti i te pūrua, ko te whārite me mātua tuhi ki te āhua x^{2}+bx=c.

$$-3x^{2}+12x=9$$

Whakawehea ngā taha e rua ki te -3.

$$\frac{-3x^{2}+12x}{-3}=\frac{9}{-3}$$

Mā te whakawehe ki te -3 ka wetekia te whakareanga ki te -3.

$$x^{2}+\frac{12}{-3}x=\frac{9}{-3}$$

Whakawehe 12 ki te -3.

$$x^{2}-4x=\frac{9}{-3}$$

Whakawehe 9 ki te -3.

$$x^{2}-4x=-3$$

Whakawehea te -4, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -2. Nā, tāpiria te pūrua o te -2 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

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

Pūrua -2.

$$x^{2}-4x+4=-3+4$$

Tāpiri -3 ki te 4.

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

Tauwehea x^{2}-4x+4. 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}.

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

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

$$\sqrt{\left(x-2\right)^{2}}=\sqrt{1}$$

Whakarūnātia.

$$x-2=1 x-2=-1$$

Me tāpiri 2 ki ngā taha e rua o te whārite.

$$x=3 x=1$$