Whakaoti i te x^2=18x+40

$${ x }^{ 2 } =18x+40$$

Whakaoti mō x

$$x=-2$$

$$x=20$$

Ngā Upane Whakamahi Tauwehe

Tangohia te 18x mai i ngā taha e rua.

$$x^{2}-18x=40$$

Tangohia te 40 mai i ngā taha e rua.

$$x^{2}-18x-40=0$$

Hei whakaoti i te whārite, whakatauwehea te x^{2}-18x-40 mā te whakamahi i te tātai x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-18 ab=-40$$

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 -40.

$$1,-40 2,-20 4,-10 5,-8$$

Tātaihia te tapeke mō ia takirua.

$$1-40=-39 2-20=-18 4-10=-6 5-8=-3$$

Ko te otinga te takirua ka hoatu i te tapeke -18.

$$a=-20 b=2$$

Me tuhi anō te kīanga whakatauwehe \left(x+a\right)\left(x+b\right) mā ngā uara i tātaihia.

$$\left(x-20\right)\left(x+2\right)$$

Hei kimi otinga whārite, me whakaoti te x-20=0 me te x+2=0.

$$x=20 x=-2$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tangohia te 18x mai i ngā taha e rua.

$$x^{2}-18x=40$$

Tangohia te 40 mai i ngā taha e rua.

$$x^{2}-18x-40=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx-40. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-18 ab=1\left(-40\right)=-40$$

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 -40.

$$1,-40 2,-20 4,-10 5,-8$$

Tātaihia te tapeke mō ia takirua.

$$1-40=-39 2-20=-18 4-10=-6 5-8=-3$$

Ko te otinga te takirua ka hoatu i te tapeke -18.

$$a=-20 b=2$$

Tuhia anō te x^{2}-18x-40 hei \left(x^{2}-20x\right)+\left(2x-40\right).

$$\left(x^{2}-20x\right)+\left(2x-40\right)$$

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

$$x\left(x-20\right)+2\left(x-20\right)$$

Whakatauwehea atu te kīanga pātahi x-20 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-20\right)\left(x+2\right)$$

Hei kimi otinga whārite, me whakaoti te x-20=0 me te x+2=0.

$$x=20 x=-2$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tangohia te 18x mai i ngā taha e rua.

$$x^{2}-18x=40$$

Tangohia te 40 mai i ngā taha e rua.

$$x^{2}-18x-40=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -18 mō b, me -40 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-18\right)±\sqrt{\left(-18\right)^{2}-4\left(-40\right)}}{2}$$

Pūrua -18.

$$x=\frac{-\left(-18\right)±\sqrt{324-4\left(-40\right)}}{2}$$

Whakareatia -4 ki te -40.

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

Tāpiri 324 ki te 160.

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

Tuhia te pūtakerua o te 484.

$$x=\frac{-\left(-18\right)±22}{2}$$

Ko te tauaro o -18 ko 18.

$$x=\frac{18±22}{2}$$

Nā, me whakaoti te whārite x=\frac{18±22}{2} ina he tāpiri te ±. Tāpiri 18 ki te 22.

$$x=\frac{40}{2}$$

Whakawehe 40 ki te 2.

$$x=20$$

Nā, me whakaoti te whārite x=\frac{18±22}{2} ina he tango te ±. Tango 22 mai i 18.

$$x=-\frac{4}{2}$$

Whakawehe -4 ki te 2.

$$x=-2$$

Kua oti te whārite te whakatau.

$$x=20 x=-2$$

Ngā Upane mō te Whakaoti i te Pūrua

Tangohia te 18x mai i ngā taha e rua.

$$x^{2}-18x=40$$

Whakawehea te -18, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -9. Nā, tāpiria te pūrua o te -9 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-18x+\left(-9\right)^{2}=40+\left(-9\right)^{2}$$

Pūrua -9.

$$x^{2}-18x+81=40+81$$

Tāpiri 40 ki te 81.

$$x^{2}-18x+81=121$$

Tauwehea x^{2}-18x+81. 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}.

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

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

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

Whakarūnātia.

$$x-9=11 x-9=-11$$

Me tāpiri 9 ki ngā taha e rua o te whārite.

$$x=20 x=-2$$