Whakaoti i te 10x^2+160=(-4x-8)^2

$$10 x ^ { 2 } + 160 = ( - 4 x - 8 ) ^ { 2 }$$

Whakaoti mō x

$$x = \frac{4}{3} = 1\frac{1}{3} \approx 1.333333333$$

$$x=-12$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(-4x-8\right)^{2}.

$$10x^{2}+160=16x^{2}+64x+64$$

Tangohia te 16x^{2} mai i ngā taha e rua.

$$10x^{2}+160-16x^{2}=64x+64$$

Pahekotia te 10x^{2} me -16x^{2}, ka -6x^{2}.

$$-6x^{2}+160=64x+64$$

Tangohia te 64x mai i ngā taha e rua.

$$-6x^{2}+160-64x=64$$

Tangohia te 64 mai i ngā taha e rua.

$$-6x^{2}+160-64x-64=0$$

Tangohia te 64 i te 160, ka 96.

$$-6x^{2}+96-64x=0$$

Whakawehea ngā taha e rua ki te 2.

$$-3x^{2}+48-32x=0$$

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

$$-3x^{2}-32x+48=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei -3x^{2}+ax+bx+48. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-32 ab=-3\times 48=-144$$

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 -144.

$$1,-144 2,-72 3,-48 4,-36 6,-24 8,-18 9,-16 12,-12$$

Tātaihia te tapeke mō ia takirua.

$$1-144=-143 2-72=-70 3-48=-45 4-36=-32 6-24=-18 8-18=-10 9-16=-7 12-12=0$$

Ko te otinga te takirua ka hoatu i te tapeke -32.

$$a=4 b=-36$$

Tuhia anō te -3x^{2}-32x+48 hei \left(-3x^{2}+4x\right)+\left(-36x+48\right).

$$\left(-3x^{2}+4x\right)+\left(-36x+48\right)$$

Tauwehea te -x i te tuatahi me te -12 i te rōpū tuarua.

$$-x\left(3x-4\right)-12\left(3x-4\right)$$

Whakatauwehea atu te kīanga pātahi 3x-4 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(3x-4\right)\left(-x-12\right)$$

Hei kimi otinga whārite, me whakaoti te 3x-4=0 me te -x-12=0.

$$x=\frac{4}{3} x=-12$$

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(-4x-8\right)^{2}.

$$10x^{2}+160=16x^{2}+64x+64$$

Tangohia te 16x^{2} mai i ngā taha e rua.

$$10x^{2}+160-16x^{2}=64x+64$$

Pahekotia te 10x^{2} me -16x^{2}, ka -6x^{2}.

$$-6x^{2}+160=64x+64$$

Tangohia te 64x mai i ngā taha e rua.

$$-6x^{2}+160-64x=64$$

Tangohia te 64 mai i ngā taha e rua.

$$-6x^{2}+160-64x-64=0$$

Tangohia te 64 i te 160, ka 96.

$$-6x^{2}+96-64x=0$$

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.

$$-6x^{2}-64x+96=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -6 mō a, -64 mō b, me 96 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-64\right)±\sqrt{\left(-64\right)^{2}-4\left(-6\right)\times 96}}{2\left(-6\right)}$$

Pūrua -64.

$$x=\frac{-\left(-64\right)±\sqrt{4096-4\left(-6\right)\times 96}}{2\left(-6\right)}$$

Whakareatia -4 ki te -6.

$$x=\frac{-\left(-64\right)±\sqrt{4096+24\times 96}}{2\left(-6\right)}$$

Whakareatia 24 ki te 96.

$$x=\frac{-\left(-64\right)±\sqrt{4096+2304}}{2\left(-6\right)}$$

Tāpiri 4096 ki te 2304.

$$x=\frac{-\left(-64\right)±\sqrt{6400}}{2\left(-6\right)}$$

Tuhia te pūtakerua o te 6400.

$$x=\frac{-\left(-64\right)±80}{2\left(-6\right)}$$

Ko te tauaro o -64 ko 64.

$$x=\frac{64±80}{2\left(-6\right)}$$

Whakareatia 2 ki te -6.

$$x=\frac{64±80}{-12}$$

Nā, me whakaoti te whārite x=\frac{64±80}{-12} ina he tāpiri te ±. Tāpiri 64 ki te 80.

$$x=\frac{144}{-12}$$

Whakawehe 144 ki te -12.

$$x=-12$$

Nā, me whakaoti te whārite x=\frac{64±80}{-12} ina he tango te ±. Tango 80 mai i 64.

$$x=-\frac{16}{-12}$$

Whakahekea te hautanga \frac{-16}{-12} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

$$x=\frac{4}{3}$$

Kua oti te whārite te whakatau.

$$x=-12 x=\frac{4}{3}$$

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(-4x-8\right)^{2}.

$$10x^{2}+160=16x^{2}+64x+64$$

Tangohia te 16x^{2} mai i ngā taha e rua.

$$10x^{2}+160-16x^{2}=64x+64$$

Pahekotia te 10x^{2} me -16x^{2}, ka -6x^{2}.

$$-6x^{2}+160=64x+64$$

Tangohia te 64x mai i ngā taha e rua.

$$-6x^{2}+160-64x=64$$

Tangohia te 160 mai i ngā taha e rua.

$$-6x^{2}-64x=64-160$$

Tangohia te 160 i te 64, ka -96.

$$-6x^{2}-64x=-96$$

Whakawehea ngā taha e rua ki te -6.

$$\frac{-6x^{2}-64x}{-6}=-\frac{96}{-6}$$

Mā te whakawehe ki te -6 ka wetekia te whakareanga ki te -6.

$$x^{2}+\left(-\frac{64}{-6}\right)x=-\frac{96}{-6}$$

Whakahekea te hautanga \frac{-64}{-6} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x^{2}+\frac{32}{3}x=-\frac{96}{-6}$$

Whakawehe -96 ki te -6.

$$x^{2}+\frac{32}{3}x=16$$

Whakawehea te \frac{32}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{16}{3}. Nā, tāpiria te pūrua o te \frac{16}{3} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{32}{3}x+\left(\frac{16}{3}\right)^{2}=16+\left(\frac{16}{3}\right)^{2}$$

Pūruatia \frac{16}{3} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{32}{3}x+\frac{256}{9}=16+\frac{256}{9}$$

Tāpiri 16 ki te \frac{256}{9}.

$$x^{2}+\frac{32}{3}x+\frac{256}{9}=\frac{400}{9}$$

Tauwehea x^{2}+\frac{32}{3}x+\frac{256}{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}.

$$\left(x+\frac{16}{3}\right)^{2}=\frac{400}{9}$$

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

$$\sqrt{\left(x+\frac{16}{3}\right)^{2}}=\sqrt{\frac{400}{9}}$$

Whakarūnātia.

$$x+\frac{16}{3}=\frac{20}{3} x+\frac{16}{3}=-\frac{20}{3}$$

Me tango \frac{16}{3} mai i ngā taha e rua o te whārite.

$$x=\frac{4}{3} x=-12$$