Whakaoti i te 3x^2+4x/2x+9=5

$$\frac{ 3 { x }^{ 2 } +4x }{ 2x+9 } = 5$$

Whakaoti mō x

$$x=-3$$

$$x=5$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tē taea kia ōrite te tāupe x ki -\frac{9}{2} nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te 2x+9.

$$3x^{2}+4x=5\left(2x+9\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 5 ki te 2x+9.

$$3x^{2}+4x=10x+45$$

Tangohia te 10x mai i ngā taha e rua.

$$3x^{2}+4x-10x=45$$

Pahekotia te 4x me -10x, ka -6x.

$$3x^{2}-6x=45$$

Tangohia te 45 mai i ngā taha e rua.

$$3x^{2}-6x-45=0$$

Whakawehea ngā taha e rua ki te 3.

$$x^{2}-2x-15=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx-15. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-2 ab=1\left(-15\right)=-15$$

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 -15.

$$1,-15 3,-5$$

Tātaihia te tapeke mō ia takirua.

$$1-15=-14 3-5=-2$$

Ko te otinga te takirua ka hoatu i te tapeke -2.

$$a=-5 b=3$$

Tuhia anō te x^{2}-2x-15 hei \left(x^{2}-5x\right)+\left(3x-15\right).

$$\left(x^{2}-5x\right)+\left(3x-15\right)$$

Tauwehea te x i te tuatahi me te 3 i te rōpū tuarua.

$$x\left(x-5\right)+3\left(x-5\right)$$

Whakatauwehea atu te kīanga pātahi x-5 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-5\right)\left(x+3\right)$$

Hei kimi otinga whārite, me whakaoti te x-5=0 me te x+3=0.

$$x=5 x=-3$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki -\frac{9}{2} nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te 2x+9.

$$3x^{2}+4x=5\left(2x+9\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 5 ki te 2x+9.

$$3x^{2}+4x=10x+45$$

Tangohia te 10x mai i ngā taha e rua.

$$3x^{2}+4x-10x=45$$

Pahekotia te 4x me -10x, ka -6x.

$$3x^{2}-6x=45$$

Tangohia te 45 mai i ngā taha e rua.

$$3x^{2}-6x-45=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 3 mō a, -6 mō b, me -45 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 3\left(-45\right)}}{2\times 3}$$

Pūrua -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\times 3\left(-45\right)}}{2\times 3}$$

Whakareatia -4 ki te 3.

$$x=\frac{-\left(-6\right)±\sqrt{36-12\left(-45\right)}}{2\times 3}$$

Whakareatia -12 ki te -45.

$$x=\frac{-\left(-6\right)±\sqrt{36+540}}{2\times 3}$$

Tāpiri 36 ki te 540.

$$x=\frac{-\left(-6\right)±\sqrt{576}}{2\times 3}$$

Tuhia te pūtakerua o te 576.

$$x=\frac{-\left(-6\right)±24}{2\times 3}$$

Ko te tauaro o -6 ko 6.

$$x=\frac{6±24}{2\times 3}$$

Whakareatia 2 ki te 3.

$$x=\frac{6±24}{6}$$

Nā, me whakaoti te whārite x=\frac{6±24}{6} ina he tāpiri te ±. Tāpiri 6 ki te 24.

$$x=\frac{30}{6}$$

Whakawehe 30 ki te 6.

$$x=5$$

Nā, me whakaoti te whārite x=\frac{6±24}{6} ina he tango te ±. Tango 24 mai i 6.

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

Whakawehe -18 ki te 6.

$$x=-3$$

Kua oti te whārite te whakatau.

$$x=5 x=-3$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki -\frac{9}{2} nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te 2x+9.

$$3x^{2}+4x=5\left(2x+9\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 5 ki te 2x+9.

$$3x^{2}+4x=10x+45$$

Tangohia te 10x mai i ngā taha e rua.

$$3x^{2}+4x-10x=45$$

Pahekotia te 4x me -10x, ka -6x.

$$3x^{2}-6x=45$$

Whakawehea ngā taha e rua ki te 3.

$$\frac{3x^{2}-6x}{3}=\frac{45}{3}$$

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

$$x^{2}+\left(-\frac{6}{3}\right)x=\frac{45}{3}$$

Whakawehe -6 ki te 3.

$$x^{2}-2x=\frac{45}{3}$$

Whakawehe 45 ki te 3.

$$x^{2}-2x=15$$

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

$$x^{2}-2x+1=15+1$$

Tāpiri 15 ki te 1.

$$x^{2}-2x+1=16$$

Tauwehea x^{2}-2x+1. 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}.

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

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

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

Whakarūnātia.

$$x-1=4 x-1=-4$$

Me tāpiri 1 ki ngā taha e rua o te whārite.

$$x=5 x=-3$$