Whakaoti i te (x-1)^2+(x+2)^2-(x-3)(x+3)=22

$$( x - 1 ) ^ { 2 } + ( x + 2 ) ^ { 2 } - ( x - 3 ) ( x + 3 ) = 22$$

Whakaoti mō x

$$x=-4$$

$$x=2$$

Ngā Upane Whakamahi Tauwehe

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(x-1\right)^{2}.

$$x^{2}-2x+1+\left(x+2\right)^{2}-\left(x-3\right)\left(x+3\right)=22$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(x+2\right)^{2}.

$$x^{2}-2x+1+x^{2}+4x+4-\left(x-3\right)\left(x+3\right)=22$$

Pahekotia te x^{2} me x^{2}, ka 2x^{2}.

$$2x^{2}-2x+1+4x+4-\left(x-3\right)\left(x+3\right)=22$$

Pahekotia te -2x me 4x, ka 2x.

$$2x^{2}+2x+1+4-\left(x-3\right)\left(x+3\right)=22$$

Tāpirihia te 1 ki te 4, ka 5.

$$2x^{2}+2x+5-\left(x-3\right)\left(x+3\right)=22$$

Whakaarohia te \left(x-3\right)\left(x+3\right). Ka taea te whakareanga te panoni ki te rerekētanga o ngā pūrua mā te ture: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Pūrua 3.

$$2x^{2}+2x+5-\left(x^{2}-9\right)=22$$

Hei kimi i te tauaro o x^{2}-9, kimihia te tauaro o ia taurangi.

$$2x^{2}+2x+5-x^{2}+9=22$$

Pahekotia te 2x^{2} me -x^{2}, ka x^{2}.

$$x^{2}+2x+5+9=22$$

Tāpirihia te 5 ki te 9, ka 14.

$$x^{2}+2x+14=22$$

Tangohia te 22 mai i ngā taha e rua.

$$x^{2}+2x+14-22=0$$

Tangohia te 22 i te 14, ka -8.

$$x^{2}+2x-8=0$$

Hei whakaoti i te whārite, whakatauwehea te x^{2}+2x-8 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.

$$a+b=2 ab=-8$$

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 -8.

$$-1,8 -2,4$$

Tātaihia te tapeke mō ia takirua.

$$-1+8=7 -2+4=2$$

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

$$a=-2 b=4$$

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

$$\left(x-2\right)\left(x+4\right)$$

Hei kimi otinga whārite, me whakaoti te x-2=0 me te x+4=0.

$$x=2 x=-4$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(x-1\right)^{2}.

$$x^{2}-2x+1+\left(x+2\right)^{2}-\left(x-3\right)\left(x+3\right)=22$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(x+2\right)^{2}.

$$x^{2}-2x+1+x^{2}+4x+4-\left(x-3\right)\left(x+3\right)=22$$

Pahekotia te x^{2} me x^{2}, ka 2x^{2}.

$$2x^{2}-2x+1+4x+4-\left(x-3\right)\left(x+3\right)=22$$

Pahekotia te -2x me 4x, ka 2x.

$$2x^{2}+2x+1+4-\left(x-3\right)\left(x+3\right)=22$$

Tāpirihia te 1 ki te 4, ka 5.

$$2x^{2}+2x+5-\left(x-3\right)\left(x+3\right)=22$$

Whakaarohia te \left(x-3\right)\left(x+3\right). Ka taea te whakareanga te panoni ki te rerekētanga o ngā pūrua mā te ture: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Pūrua 3.

$$2x^{2}+2x+5-\left(x^{2}-9\right)=22$$

Hei kimi i te tauaro o x^{2}-9, kimihia te tauaro o ia taurangi.

$$2x^{2}+2x+5-x^{2}+9=22$$

Pahekotia te 2x^{2} me -x^{2}, ka x^{2}.

$$x^{2}+2x+5+9=22$$

Tāpirihia te 5 ki te 9, ka 14.

$$x^{2}+2x+14=22$$

Tangohia te 22 mai i ngā taha e rua.

$$x^{2}+2x+14-22=0$$

Tangohia te 22 i te 14, ka -8.

$$x^{2}+2x-8=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx-8. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=2 ab=1\left(-8\right)=-8$$

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 -8.

$$-1,8 -2,4$$

Tātaihia te tapeke mō ia takirua.

$$-1+8=7 -2+4=2$$

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

$$a=-2 b=4$$

Tuhia anō te x^{2}+2x-8 hei \left(x^{2}-2x\right)+\left(4x-8\right).

$$\left(x^{2}-2x\right)+\left(4x-8\right)$$

Tauwehea te x i te tuatahi me te 4 i te rōpū tuarua.

$$x\left(x-2\right)+4\left(x-2\right)$$

Whakatauwehea atu te kīanga pātahi x-2 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-2\right)\left(x+4\right)$$

Hei kimi otinga whārite, me whakaoti te x-2=0 me te x+4=0.

$$x=2 x=-4$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(x-1\right)^{2}.

$$x^{2}-2x+1+\left(x+2\right)^{2}-\left(x-3\right)\left(x+3\right)=22$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(x+2\right)^{2}.

$$x^{2}-2x+1+x^{2}+4x+4-\left(x-3\right)\left(x+3\right)=22$$

Pahekotia te x^{2} me x^{2}, ka 2x^{2}.

$$2x^{2}-2x+1+4x+4-\left(x-3\right)\left(x+3\right)=22$$

Pahekotia te -2x me 4x, ka 2x.

$$2x^{2}+2x+1+4-\left(x-3\right)\left(x+3\right)=22$$

Tāpirihia te 1 ki te 4, ka 5.

$$2x^{2}+2x+5-\left(x-3\right)\left(x+3\right)=22$$

Whakaarohia te \left(x-3\right)\left(x+3\right). Ka taea te whakareanga te panoni ki te rerekētanga o ngā pūrua mā te ture: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Pūrua 3.

$$2x^{2}+2x+5-\left(x^{2}-9\right)=22$$

Hei kimi i te tauaro o x^{2}-9, kimihia te tauaro o ia taurangi.

$$2x^{2}+2x+5-x^{2}+9=22$$

Pahekotia te 2x^{2} me -x^{2}, ka x^{2}.

$$x^{2}+2x+5+9=22$$

Tāpirihia te 5 ki te 9, ka 14.

$$x^{2}+2x+14=22$$

Tangohia te 22 mai i ngā taha e rua.

$$x^{2}+2x+14-22=0$$

Tangohia te 22 i te 14, ka -8.

$$x^{2}+2x-8=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 2 mō b, me -8 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2±\sqrt{2^{2}-4\left(-8\right)}}{2}$$

Pūrua 2.

$$x=\frac{-2±\sqrt{4-4\left(-8\right)}}{2}$$

Whakareatia -4 ki te -8.

$$x=\frac{-2±\sqrt{4+32}}{2}$$

Tāpiri 4 ki te 32.

$$x=\frac{-2±\sqrt{36}}{2}$$

Tuhia te pūtakerua o te 36.

$$x=\frac{-2±6}{2}$$

Nā, me whakaoti te whārite x=\frac{-2±6}{2} ina he tāpiri te ±. Tāpiri -2 ki te 6.

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

Whakawehe 4 ki te 2.

$$x=2$$

Nā, me whakaoti te whārite x=\frac{-2±6}{2} ina he tango te ±. Tango 6 mai i -2.

$$x=-\frac{8}{2}$$

Whakawehe -8 ki te 2.

$$x=-4$$

Kua oti te whārite te whakatau.

$$x=2 x=-4$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(x-1\right)^{2}.

$$x^{2}-2x+1+\left(x+2\right)^{2}-\left(x-3\right)\left(x+3\right)=22$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(x+2\right)^{2}.

$$x^{2}-2x+1+x^{2}+4x+4-\left(x-3\right)\left(x+3\right)=22$$

Pahekotia te x^{2} me x^{2}, ka 2x^{2}.

$$2x^{2}-2x+1+4x+4-\left(x-3\right)\left(x+3\right)=22$$

Pahekotia te -2x me 4x, ka 2x.

$$2x^{2}+2x+1+4-\left(x-3\right)\left(x+3\right)=22$$

Tāpirihia te 1 ki te 4, ka 5.

$$2x^{2}+2x+5-\left(x-3\right)\left(x+3\right)=22$$

Whakaarohia te \left(x-3\right)\left(x+3\right). Ka taea te whakareanga te panoni ki te rerekētanga o ngā pūrua mā te ture: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Pūrua 3.

$$2x^{2}+2x+5-\left(x^{2}-9\right)=22$$

Hei kimi i te tauaro o x^{2}-9, kimihia te tauaro o ia taurangi.

$$2x^{2}+2x+5-x^{2}+9=22$$

Pahekotia te 2x^{2} me -x^{2}, ka x^{2}.

$$x^{2}+2x+5+9=22$$

Tāpirihia te 5 ki te 9, ka 14.

$$x^{2}+2x+14=22$$

Tangohia te 14 mai i ngā taha e rua.

$$x^{2}+2x=22-14$$

Tangohia te 14 i te 22, ka 8.

$$x^{2}+2x=8$$

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.

$$x^{2}+2x+1^{2}=8+1^{2}$$

Pūrua 1.

$$x^{2}+2x+1=8+1$$

Tāpiri 8 ki te 1.

$$x^{2}+2x+1=9$$

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}.

$$\left(x+1\right)^{2}=9$$

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

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{9}$$

Whakarūnātia.

$$x+1=3 x+1=-3$$

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

$$x=2 x=-4$$