Whakaoti i te x^2=2x+48

$$x ^ { 2 } = 2 x + 48$$

Whakaoti mō x

$$x=-6$$

$$x=8$$

Ngā Upane Whakamahi Tauwehe

Tangohia te 2x mai i ngā taha e rua.

$$x^{2}-2x=48$$

Tangohia te 48 mai i ngā taha e rua.

$$x^{2}-2x-48=0$$

Hei whakaoti i te whārite, whakatauwehea te x^{2}-2x-48 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.

$$a+b=-2 ab=-48$$

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 -48.

$$1,-48 2,-24 3,-16 4,-12 6,-8$$

Tātaihia te tapeke mō ia takirua.

$$1-48=-47 2-24=-22 3-16=-13 4-12=-8 6-8=-2$$

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

$$a=-8 b=6$$

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

$$\left(x-8\right)\left(x+6\right)$$

Hei kimi otinga whārite, me whakaoti te x-8=0 me te x+6=0.

$$x=8 x=-6$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tangohia te 2x mai i ngā taha e rua.

$$x^{2}-2x=48$$

Tangohia te 48 mai i ngā taha e rua.

$$x^{2}-2x-48=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx-48. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-2 ab=1\left(-48\right)=-48$$

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 -48.

$$1,-48 2,-24 3,-16 4,-12 6,-8$$

Tātaihia te tapeke mō ia takirua.

$$1-48=-47 2-24=-22 3-16=-13 4-12=-8 6-8=-2$$

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

$$a=-8 b=6$$

Tuhia anō te x^{2}-2x-48 hei \left(x^{2}-8x\right)+\left(6x-48\right).

$$\left(x^{2}-8x\right)+\left(6x-48\right)$$

Tauwehea te x i te tuatahi me te 6 i te rōpū tuarua.

$$x\left(x-8\right)+6\left(x-8\right)$$

Whakatauwehea atu te kīanga pātahi x-8 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-8\right)\left(x+6\right)$$

Hei kimi otinga whārite, me whakaoti te x-8=0 me te x+6=0.

$$x=8 x=-6$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tangohia te 2x mai i ngā taha e rua.

$$x^{2}-2x=48$$

Tangohia te 48 mai i ngā taha e rua.

$$x^{2}-2x-48=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -2 mō b, me -48 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\left(-48\right)}}{2}$$

Pūrua -2.

$$x=\frac{-\left(-2\right)±\sqrt{4-4\left(-48\right)}}{2}$$

Whakareatia -4 ki te -48.

$$x=\frac{-\left(-2\right)±\sqrt{4+192}}{2}$$

Tāpiri 4 ki te 192.

$$x=\frac{-\left(-2\right)±\sqrt{196}}{2}$$

Tuhia te pūtakerua o te 196.

$$x=\frac{-\left(-2\right)±14}{2}$$

Ko te tauaro o -2 ko 2.

$$x=\frac{2±14}{2}$$

Nā, me whakaoti te whārite x=\frac{2±14}{2} ina he tāpiri te ±. Tāpiri 2 ki te 14.

$$x=\frac{16}{2}$$

Whakawehe 16 ki te 2.

$$x=8$$

Nā, me whakaoti te whārite x=\frac{2±14}{2} ina he tango te ±. Tango 14 mai i 2.

$$x=-\frac{12}{2}$$

Whakawehe -12 ki te 2.

$$x=-6$$

Kua oti te whārite te whakatau.

$$x=8 x=-6$$

Ngā Upane mō te Whakaoti i te Pūrua

Tangohia te 2x mai i ngā taha e rua.

$$x^{2}-2x=48$$

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.

$$x^{2}-2x+1=48+1$$

Tāpiri 48 ki te 1.

$$x^{2}-2x+1=49$$

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}.

$$\left(x-1\right)^{2}=49$$

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

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{49}$$

Whakarūnātia.

$$x-1=7 x-1=-7$$

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

$$x=8 x=-6$$