Whakaoti i te 2x^2=18x+20

$$2 x ^ { 2 } = 18 x + 20$$

Whakaoti mō x

$$x=-1$$

$$x=10$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tangohia te 18x mai i ngā taha e rua.

$$2x^{2}-18x=20$$

Tangohia te 20 mai i ngā taha e rua.

$$2x^{2}-18x-20=0$$

Whakawehea ngā taha e rua ki te 2.

$$x^{2}-9x-10=0$$

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

$$a+b=-9 ab=1\left(-10\right)=-10$$

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 -9.

$$a=-10 b=1$$

Tuhia anō te x^{2}-9x-10 hei \left(x^{2}-10x\right)+\left(x-10\right).

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

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

$$x\left(x-10\right)+x-10$$

Whakatauwehea atu te kīanga pātahi x-10 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-10\right)\left(x+1\right)$$

Hei kimi otinga whārite, me whakaoti te x-10=0 me te x+1=0.

$$x=10 x=-1$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tangohia te 18x mai i ngā taha e rua.

$$2x^{2}-18x=20$$

Tangohia te 20 mai i ngā taha e rua.

$$2x^{2}-18x-20=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 2 mō a, -18 mō b, me -20 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-18\right)±\sqrt{\left(-18\right)^{2}-4\times 2\left(-20\right)}}{2\times 2}$$

Pūrua -18.

$$x=\frac{-\left(-18\right)±\sqrt{324-4\times 2\left(-20\right)}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$x=\frac{-\left(-18\right)±\sqrt{324-8\left(-20\right)}}{2\times 2}$$

Whakareatia -8 ki te -20.

$$x=\frac{-\left(-18\right)±\sqrt{324+160}}{2\times 2}$$

Tāpiri 324 ki te 160.

$$x=\frac{-\left(-18\right)±\sqrt{484}}{2\times 2}$$

Tuhia te pūtakerua o te 484.

$$x=\frac{-\left(-18\right)±22}{2\times 2}$$

Ko te tauaro o -18 ko 18.

$$x=\frac{18±22}{2\times 2}$$

Whakareatia 2 ki te 2.

$$x=\frac{18±22}{4}$$

Nā, me whakaoti te whārite x=\frac{18±22}{4} ina he tāpiri te ±. Tāpiri 18 ki te 22.

$$x=\frac{40}{4}$$

Whakawehe 40 ki te 4.

$$x=10$$

Nā, me whakaoti te whārite x=\frac{18±22}{4} ina he tango te ±. Tango 22 mai i 18.

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

Whakawehe -4 ki te 4.

$$x=-1$$

Kua oti te whārite te whakatau.

$$x=10 x=-1$$

Ngā Upane mō te Whakaoti i te Pūrua

Tangohia te 18x mai i ngā taha e rua.

$$2x^{2}-18x=20$$

Whakawehea ngā taha e rua ki te 2.

$$\frac{2x^{2}-18x}{2}=\frac{20}{2}$$

Mā te whakawehe ki te 2 ka wetekia te whakareanga ki te 2.

$$x^{2}+\left(-\frac{18}{2}\right)x=\frac{20}{2}$$

Whakawehe -18 ki te 2.

$$x^{2}-9x=\frac{20}{2}$$

Whakawehe 20 ki te 2.

$$x^{2}-9x=10$$

Whakawehea te -9, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{9}{2}. Nā, tāpiria te pūrua o te -\frac{9}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-9x+\left(-\frac{9}{2}\right)^{2}=10+\left(-\frac{9}{2}\right)^{2}$$

Pūruatia -\frac{9}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-9x+\frac{81}{4}=10+\frac{81}{4}$$

Tāpiri 10 ki te \frac{81}{4}.

$$x^{2}-9x+\frac{81}{4}=\frac{121}{4}$$

Tauwehea x^{2}-9x+\frac{81}{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}.

$$\left(x-\frac{9}{2}\right)^{2}=\frac{121}{4}$$

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

$$\sqrt{\left(x-\frac{9}{2}\right)^{2}}=\sqrt{\frac{121}{4}}$$

Whakarūnātia.

$$x-\frac{9}{2}=\frac{11}{2} x-\frac{9}{2}=-\frac{11}{2}$$

Me tāpiri \frac{9}{2} ki ngā taha e rua o te whārite.

$$x=10 x=-1$$