Whakaoti i te (v+4)^2=2v^2+2v+9

$$( v + 4 ) ^ { 2 } = 2 v ^ { 2 } + 2 v + 9$$

Whakaoti mō v

$$v=-1$$

$$v=7$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(v+4\right)^{2}.

$$v^{2}+8v+16=2v^{2}+2v+9$$

Tangohia te 2v^{2} mai i ngā taha e rua.

$$v^{2}+8v+16-2v^{2}=2v+9$$

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

$$-v^{2}+8v+16=2v+9$$

Tangohia te 2v mai i ngā taha e rua.

$$-v^{2}+8v+16-2v=9$$

Pahekotia te 8v me -2v, ka 6v.

$$-v^{2}+6v+16=9$$

Tangohia te 9 mai i ngā taha e rua.

$$-v^{2}+6v+16-9=0$$

Tangohia te 9 i te 16, ka 7.

$$-v^{2}+6v+7=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei -v^{2}+av+bv+7. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=6 ab=-7=-7$$

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. Ko te takirua anake pērā ko te otinga pūnaha.

$$a=7 b=-1$$

Tuhia anō te -v^{2}+6v+7 hei \left(-v^{2}+7v\right)+\left(-v+7\right).

$$\left(-v^{2}+7v\right)+\left(-v+7\right)$$

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

$$-v\left(v-7\right)-\left(v-7\right)$$

Whakatauwehea atu te kīanga pātahi v-7 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(v-7\right)\left(-v-1\right)$$

Hei kimi otinga whārite, me whakaoti te v-7=0 me te -v-1=0.

$$v=7 v=-1$$

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(v+4\right)^{2}.

$$v^{2}+8v+16=2v^{2}+2v+9$$

Tangohia te 2v^{2} mai i ngā taha e rua.

$$v^{2}+8v+16-2v^{2}=2v+9$$

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

$$-v^{2}+8v+16=2v+9$$

Tangohia te 2v mai i ngā taha e rua.

$$-v^{2}+8v+16-2v=9$$

Pahekotia te 8v me -2v, ka 6v.

$$-v^{2}+6v+16=9$$

Tangohia te 9 mai i ngā taha e rua.

$$-v^{2}+6v+16-9=0$$

Tangohia te 9 i te 16, ka 7.

$$-v^{2}+6v+7=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -1 mō a, 6 mō b, me 7 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$v=\frac{-6±\sqrt{6^{2}-4\left(-1\right)\times 7}}{2\left(-1\right)}$$

Pūrua 6.

$$v=\frac{-6±\sqrt{36-4\left(-1\right)\times 7}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$v=\frac{-6±\sqrt{36+4\times 7}}{2\left(-1\right)}$$

Whakareatia 4 ki te 7.

$$v=\frac{-6±\sqrt{36+28}}{2\left(-1\right)}$$

Tāpiri 36 ki te 28.

$$v=\frac{-6±\sqrt{64}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te 64.

$$v=\frac{-6±8}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$v=\frac{-6±8}{-2}$$

Nā, me whakaoti te whārite v=\frac{-6±8}{-2} ina he tāpiri te ±. Tāpiri -6 ki te 8.

$$v=\frac{2}{-2}$$

Whakawehe 2 ki te -2.

$$v=-1$$

Nā, me whakaoti te whārite v=\frac{-6±8}{-2} ina he tango te ±. Tango 8 mai i -6.

$$v=-\frac{14}{-2}$$

Whakawehe -14 ki te -2.

$$v=7$$

Kua oti te whārite te whakatau.

$$v=-1 v=7$$

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(v+4\right)^{2}.

$$v^{2}+8v+16=2v^{2}+2v+9$$

Tangohia te 2v^{2} mai i ngā taha e rua.

$$v^{2}+8v+16-2v^{2}=2v+9$$

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

$$-v^{2}+8v+16=2v+9$$

Tangohia te 2v mai i ngā taha e rua.

$$-v^{2}+8v+16-2v=9$$

Pahekotia te 8v me -2v, ka 6v.

$$-v^{2}+6v+16=9$$

Tangohia te 16 mai i ngā taha e rua.

$$-v^{2}+6v=9-16$$

Tangohia te 16 i te 9, ka -7.

$$-v^{2}+6v=-7$$

Whakawehea ngā taha e rua ki te -1.

$$\frac{-v^{2}+6v}{-1}=-\frac{7}{-1}$$

Mā te whakawehe ki te -1 ka wetekia te whakareanga ki te -1.

$$v^{2}+\frac{6}{-1}v=-\frac{7}{-1}$$

Whakawehe 6 ki te -1.

$$v^{2}-6v=-\frac{7}{-1}$$

Whakawehe -7 ki te -1.

$$v^{2}-6v=7$$

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.

$$v^{2}-6v+\left(-3\right)^{2}=7+\left(-3\right)^{2}$$

Pūrua -3.

$$v^{2}-6v+9=7+9$$

Tāpiri 7 ki te 9.

$$v^{2}-6v+9=16$$

Tauwehea te v^{2}-6v+9. Ko te tikanga, ina ko x^{2}+bx+c he pūrua tika, ka taea te tauwehe i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

$$\left(v-3\right)^{2}=16$$

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

$$\sqrt{\left(v-3\right)^{2}}=\sqrt{16}$$

Whakarūnātia.

$$v-3=4 v-3=-4$$

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

$$v=7 v=-1$$