Whakaoti i te 1-2x+x^2/2=17

$$1-2x+ \frac{ { x }^{ 2 } }{ 2 } =17$$

Whakaoti mō x

$$x=-4$$

$$x=8$$

Ngā Upane Whakamahi Tauwehe

Whakareatia ngā taha e rua o te whārite ki te 2.

$$2-4x+x^{2}=34$$

Tangohia te 34 mai i ngā taha e rua.

$$2-4x+x^{2}-34=0$$

Tangohia te 34 i te 2, ka -32.

$$-32-4x+x^{2}=0$$

Hurinahatia te pūrau ki te āhua tānga ngahuru. Whakaraupapahia ngā kīanga tau mai i te pū teitei rawa ki te mea iti rawa.

$$x^{2}-4x-32=0$$

Hei whakaoti i te whārite, whakatauwehea te x^{2}-4x-32 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.

$$a+b=-4 ab=-32$$

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 -32.

$$1,-32 2,-16 4,-8$$

Tātaihia te tapeke mō ia takirua.

$$1-32=-31 2-16=-14 4-8=-4$$

Ko te otinga te takirua ka hoatu i te tapeke -4.

$$a=-8 b=4$$

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

$$\left(x-8\right)\left(x+4\right)$$

Hei kimi otinga whārite, me whakaoti te x-8=0 me te x+4=0.

$$x=8 x=-4$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakareatia ngā taha e rua o te whārite ki te 2.

$$2-4x+x^{2}=34$$

Tangohia te 34 mai i ngā taha e rua.

$$2-4x+x^{2}-34=0$$

Tangohia te 34 i te 2, ka -32.

$$-32-4x+x^{2}=0$$

Hurinahatia te pūrau ki te āhua tānga ngahuru. Whakaraupapahia ngā kīanga tau mai i te pū teitei rawa ki te mea iti rawa.

$$x^{2}-4x-32=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx-32. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-4 ab=1\left(-32\right)=-32$$

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 -32.

$$1,-32 2,-16 4,-8$$

Tātaihia te tapeke mō ia takirua.

$$1-32=-31 2-16=-14 4-8=-4$$

Ko te otinga te takirua ka hoatu i te tapeke -4.

$$a=-8 b=4$$

Tuhia anō te x^{2}-4x-32 hei \left(x^{2}-8x\right)+\left(4x-32\right).

$$\left(x^{2}-8x\right)+\left(4x-32\right)$$

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

$$x\left(x-8\right)+4\left(x-8\right)$$

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

$$\left(x-8\right)\left(x+4\right)$$

Hei kimi otinga whārite, me whakaoti te x-8=0 me te x+4=0.

$$x=8 x=-4$$

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.

$$\frac{1}{2}x^{2}-2x+1=17$$

Me tango 17 mai i ngā taha e rua o te whārite.

$$\frac{1}{2}x^{2}-2x+1-17=17-17$$

Mā te tango i te 17 i a ia ake anō ka toe ko te 0.

$$\frac{1}{2}x^{2}-2x+1-17=0$$

Tango 17 mai i 1.

$$\frac{1}{2}x^{2}-2x-16=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi \frac{1}{2} mō a, -2 mō b, me -16 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\times \frac{1}{2}\left(-16\right)}}{2\times \frac{1}{2}}$$

Pūrua -2.

$$x=\frac{-\left(-2\right)±\sqrt{4-4\times \frac{1}{2}\left(-16\right)}}{2\times \frac{1}{2}}$$

Whakareatia -4 ki te \frac{1}{2}.

$$x=\frac{-\left(-2\right)±\sqrt{4-2\left(-16\right)}}{2\times \frac{1}{2}}$$

Whakareatia -2 ki te -16.

$$x=\frac{-\left(-2\right)±\sqrt{4+32}}{2\times \frac{1}{2}}$$

Tāpiri 4 ki te 32.

$$x=\frac{-\left(-2\right)±\sqrt{36}}{2\times \frac{1}{2}}$$

Tuhia te pūtakerua o te 36.

$$x=\frac{-\left(-2\right)±6}{2\times \frac{1}{2}}$$

Ko te tauaro o -2 ko 2.

$$x=\frac{2±6}{2\times \frac{1}{2}}$$

Whakareatia 2 ki te \frac{1}{2}.

$$x=\frac{2±6}{1}$$

Nā, me whakaoti te whārite x=\frac{2±6}{1} ina he tāpiri te ±. Tāpiri 2 ki te 6.

$$x=\frac{8}{1}$$

Whakawehe 8 ki te 1.

$$x=8$$

Nā, me whakaoti te whārite x=\frac{2±6}{1} ina he tango te ±. Tango 6 mai i 2.

$$x=-\frac{4}{1}$$

Whakawehe -4 ki te 1.

$$x=-4$$

Kua oti te whārite te whakatau.

$$x=8 x=-4$$

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.

$$\frac{1}{2}x^{2}-2x+1=17$$

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

$$\frac{1}{2}x^{2}-2x+1-1=17-1$$

Mā te tango i te 1 i a ia ake anō ka toe ko te 0.

$$\frac{1}{2}x^{2}-2x=17-1$$

Tango 1 mai i 17.

$$\frac{1}{2}x^{2}-2x=16$$

Me whakarea ngā taha e rua ki te 2.

$$\frac{\frac{1}{2}x^{2}-2x}{\frac{1}{2}}=\frac{16}{\frac{1}{2}}$$

Mā te whakawehe ki te \frac{1}{2} ka wetekia te whakareanga ki te \frac{1}{2}.

$$x^{2}+\left(-\frac{2}{\frac{1}{2}}\right)x=\frac{16}{\frac{1}{2}}$$

Whakawehe -2 ki te \frac{1}{2} mā te whakarea -2 ki te tau huripoki o \frac{1}{2}.

$$x^{2}-4x=\frac{16}{\frac{1}{2}}$$

Whakawehe 16 ki te \frac{1}{2} mā te whakarea 16 ki te tau huripoki o \frac{1}{2}.

$$x^{2}-4x=32$$

Whakawehea te -4, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -2. Nā, tāpiria te pūrua o te -2 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-4x+\left(-2\right)^{2}=32+\left(-2\right)^{2}$$

Pūrua -2.

$$x^{2}-4x+4=32+4$$

Tāpiri 32 ki te 4.

$$x^{2}-4x+4=36$$

Tauwehea x^{2}-4x+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}.

$$\left(x-2\right)^{2}=36$$

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

$$\sqrt{\left(x-2\right)^{2}}=\sqrt{36}$$

Whakarūnātia.

$$x-2=6 x-2=-6$$

Me tāpiri 2 ki ngā taha e rua o te whārite.

$$x=8 x=-4$$