Whakaoti i te 2(x-3)^2+6=14

$$2 ( x - 3 ) ^ { 2 } + 6 = 14$$

Whakaoti mō x

$$x=5$$

$$x=1$$

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-3\right)^{2}.

$$2\left(x^{2}-6x+9\right)+6=14$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te x^{2}-6x+9.

$$2x^{2}-12x+18+6=14$$

Tāpirihia te 18 ki te 6, ka 24.

$$2x^{2}-12x+24=14$$

Tangohia te 14 mai i ngā taha e rua.

$$2x^{2}-12x+24-14=0$$

Tangohia te 14 i te 24, ka 10.

$$2x^{2}-12x+10=0$$

Whakawehea ngā taha e rua ki te 2.

$$x^{2}-6x+5=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx+5. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-6 ab=1\times 5=5$$

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. Ko te takirua anake pērā ko te otinga pūnaha.

$$a=-5 b=-1$$

Tuhia anō te x^{2}-6x+5 hei \left(x^{2}-5x\right)+\left(-x+5\right).

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

Tauwehea te x i te tuatahi me te -1 i te rōpū tuarua.

$$x\left(x-5\right)-\left(x-5\right)$$

Whakatauwehea atu te kīanga pātahi x-5 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-5\right)\left(x-1\right)$$

Hei kimi otinga whārite, me whakaoti te x-5=0 me te x-1=0.

$$x=5 x=1$$

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-3\right)^{2}.

$$2\left(x^{2}-6x+9\right)+6=14$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te x^{2}-6x+9.

$$2x^{2}-12x+18+6=14$$

Tāpirihia te 18 ki te 6, ka 24.

$$2x^{2}-12x+24=14$$

Tangohia te 14 mai i ngā taha e rua.

$$2x^{2}-12x+24-14=0$$

Tangohia te 14 i te 24, ka 10.

$$2x^{2}-12x+10=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 2 mō a, -12 mō b, me 10 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\times 2\times 10}}{2\times 2}$$

Pūrua -12.

$$x=\frac{-\left(-12\right)±\sqrt{144-4\times 2\times 10}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$x=\frac{-\left(-12\right)±\sqrt{144-8\times 10}}{2\times 2}$$

Whakareatia -8 ki te 10.

$$x=\frac{-\left(-12\right)±\sqrt{144-80}}{2\times 2}$$

Tāpiri 144 ki te -80.

$$x=\frac{-\left(-12\right)±\sqrt{64}}{2\times 2}$$

Tuhia te pūtakerua o te 64.

$$x=\frac{-\left(-12\right)±8}{2\times 2}$$

Ko te tauaro o -12 ko 12.

$$x=\frac{12±8}{2\times 2}$$

Whakareatia 2 ki te 2.

$$x=\frac{12±8}{4}$$

Nā, me whakaoti te whārite x=\frac{12±8}{4} ina he tāpiri te ±. Tāpiri 12 ki te 8.

$$x=\frac{20}{4}$$

Whakawehe 20 ki te 4.

$$x=5$$

Nā, me whakaoti te whārite x=\frac{12±8}{4} ina he tango te ±. Tango 8 mai i 12.

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

Whakawehe 4 ki te 4.

$$x=1$$

Kua oti te whārite te whakatau.

$$x=5 x=1$$

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-3\right)^{2}.

$$2\left(x^{2}-6x+9\right)+6=14$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te x^{2}-6x+9.

$$2x^{2}-12x+18+6=14$$

Tāpirihia te 18 ki te 6, ka 24.

$$2x^{2}-12x+24=14$$

Tangohia te 24 mai i ngā taha e rua.

$$2x^{2}-12x=14-24$$

Tangohia te 24 i te 14, ka -10.

$$2x^{2}-12x=-10$$

Whakawehea ngā taha e rua ki te 2.

$$\frac{2x^{2}-12x}{2}=-\frac{10}{2}$$

Mā te whakawehe ki te 2 ka wetekia te whakareanga ki te 2.

$$x^{2}+\left(-\frac{12}{2}\right)x=-\frac{10}{2}$$

Whakawehe -12 ki te 2.

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

Whakawehe -10 ki te 2.

$$x^{2}-6x=-5$$

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.

$$x^{2}-6x+\left(-3\right)^{2}=-5+\left(-3\right)^{2}$$

Pūrua -3.

$$x^{2}-6x+9=-5+9$$

Tāpiri -5 ki te 9.

$$x^{2}-6x+9=4$$

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}.

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

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

$$\sqrt{\left(x-3\right)^{2}}=\sqrt{4}$$

Whakarūnātia.

$$x-3=2 x-3=-2$$

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

$$x=5 x=1$$