Whakaoti i te x^2+left(-3x+10right)^2=20

$${ x }^{ 2 } + { \left(-3x+10 \right) }^{ 2 } =20$$

Whakaoti mō x

$$x=2$$

$$x=4$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(-3x+10\right)^{2}.

$$x^{2}+9x^{2}-60x+100=20$$

Pahekotia te x^{2} me 9x^{2}, ka 10x^{2}.

$$10x^{2}-60x+100=20$$

Tangohia te 20 mai i ngā taha e rua.

$$10x^{2}-60x+100-20=0$$

Tangohia te 20 i te 100, ka 80.

$$10x^{2}-60x+80=0$$

Whakawehea ngā taha e rua ki te 10.

$$x^{2}-6x+8=0$$

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.

$$a+b=-6 ab=1\times 8=8$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōraro te a+b, he tōraro hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 8.

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

Tātaihia te tapeke mō ia takirua.

$$-1-8=-9 -2-4=-6$$

Ko te otinga te takirua ka hoatu i te tapeke -6.

$$a=-4 b=-2$$

Tuhia anō te x^{2}-6x+8 hei \left(x^{2}-4x\right)+\left(-2x+8\right).

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

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

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

Whakatauwehea atu te kīanga pātahi x-4 mā te whakamahi i te āhuatanga tātai tohatoha.

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

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

$$x=4 x=2$$

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(-3x+10\right)^{2}.

$$x^{2}+9x^{2}-60x+100=20$$

Pahekotia te x^{2} me 9x^{2}, ka 10x^{2}.

$$10x^{2}-60x+100=20$$

Tangohia te 20 mai i ngā taha e rua.

$$10x^{2}-60x+100-20=0$$

Tangohia te 20 i te 100, ka 80.

$$10x^{2}-60x+80=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 10 mō a, -60 mō b, me 80 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-60\right)±\sqrt{\left(-60\right)^{2}-4\times 10\times 80}}{2\times 10}$$

Pūrua -60.

$$x=\frac{-\left(-60\right)±\sqrt{3600-4\times 10\times 80}}{2\times 10}$$

Whakareatia -4 ki te 10.

$$x=\frac{-\left(-60\right)±\sqrt{3600-40\times 80}}{2\times 10}$$

Whakareatia -40 ki te 80.

$$x=\frac{-\left(-60\right)±\sqrt{3600-3200}}{2\times 10}$$

Tāpiri 3600 ki te -3200.

$$x=\frac{-\left(-60\right)±\sqrt{400}}{2\times 10}$$

Tuhia te pūtakerua o te 400.

$$x=\frac{-\left(-60\right)±20}{2\times 10}$$

Ko te tauaro o -60 ko 60.

$$x=\frac{60±20}{2\times 10}$$

Whakareatia 2 ki te 10.

$$x=\frac{60±20}{20}$$

Nā, me whakaoti te whārite x=\frac{60±20}{20} ina he tāpiri te ±. Tāpiri 60 ki te 20.

$$x=\frac{80}{20}$$

Whakawehe 80 ki te 20.

$$x=4$$

Nā, me whakaoti te whārite x=\frac{60±20}{20} ina he tango te ±. Tango 20 mai i 60.

$$x=\frac{40}{20}$$

Whakawehe 40 ki te 20.

$$x=2$$

Kua oti te whārite te whakatau.

$$x=4 x=2$$

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(-3x+10\right)^{2}.

$$x^{2}+9x^{2}-60x+100=20$$

Pahekotia te x^{2} me 9x^{2}, ka 10x^{2}.

$$10x^{2}-60x+100=20$$

Tangohia te 100 mai i ngā taha e rua.

$$10x^{2}-60x=20-100$$

Tangohia te 100 i te 20, ka -80.

$$10x^{2}-60x=-80$$

Whakawehea ngā taha e rua ki te 10.

$$\frac{10x^{2}-60x}{10}=-\frac{80}{10}$$

Mā te whakawehe ki te 10 ka wetekia te whakareanga ki te 10.

$$x^{2}+\left(-\frac{60}{10}\right)x=-\frac{80}{10}$$

Whakawehe -60 ki te 10.

$$x^{2}-6x=-\frac{80}{10}$$

Whakawehe -80 ki te 10.

$$x^{2}-6x=-8$$

Whakawehea te -6, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -3. Nā, tāpiria te pūrua o te -3 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-6x+\left(-3\right)^{2}=-8+\left(-3\right)^{2}$$

Pūrua -3.

$$x^{2}-6x+9=-8+9$$

Tāpiri -8 ki te 9.

$$x^{2}-6x+9=1$$

Tauwehea x^{2}-6x+9. 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}.

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

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

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

Whakarūnātia.

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

Me tāpiri 3 ki ngā taha e rua o te whārite.

$$x=4 x=2$$