Whakaoti i te x^2+2x=48

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

Whakaoti mō x

$$x=-8$$

$$x=6$$

Ngā Upane Whakamahi Tauwehe

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 tōrunga te a+b, he nui ake te uara pū o te tau tōrunga i tō te tōraro. 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.

$$a=-6 b=8$$

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

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

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

$$x=6 x=-8$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

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 tōrunga te a+b, he nui ake te uara pū o te tau tōrunga i tō te tōraro. 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.

$$a=-6 b=8$$

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

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

Tauwehea te x i te tuatahi me te 8 i te rōpū tuarua.

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

Whakatauwehea atu te kīanga pātahi x-6 mā te whakamahi i te āhuatanga tātai tohatoha.

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

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

$$x=6 x=-8$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

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.

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

Me tango 48 mai i ngā taha e rua o te whārite.

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

Mā te tango i te 48 i a ia ake anō ka toe ko te 0.

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

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}.

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

Pūrua 2.

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

Whakareatia -4 ki te -48.

$$x=\frac{-2±\sqrt{4+192}}{2}$$

Tāpiri 4 ki te 192.

$$x=\frac{-2±\sqrt{196}}{2}$$

Tuhia te pūtakerua o te 196.

$$x=\frac{-2±14}{2}$$

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

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

Whakawehe 12 ki te 2.

$$x=6$$

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

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

Whakawehe -16 ki te 2.

$$x=-8$$

Kua oti te whārite te whakatau.

$$x=6 x=-8$$

Ngā Upane mō te Whakaoti i te Pūrua

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.

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

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.

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

Pūrua 1.

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

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 tango 1 mai i ngā taha e rua o te whārite.

$$x=6 x=-8$$