Whakaoti i te (4sqrt{3}+frac{xsqrt{3}}{2})^2+x^2/4=156

$$( 4 \sqrt { 3 } + \frac { x \sqrt { 3 } } { 2 } ) ^ { 2 } + \frac { x ^ { 2 } } { 4 } = 156$$

Whakaoti mō x

$$x=-18$$

$$x=6$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakareatia ngā taha e rua o te whārite ki te 4.

$$4\left(4\sqrt{3}+\frac{x\sqrt{3}}{2}\right)^{2}+x^{2}=624$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(4\sqrt{3}+\frac{x\sqrt{3}}{2}\right)^{2}.

$$4\left(16\left(\sqrt{3}\right)^{2}+8\sqrt{3}\times \frac{x\sqrt{3}}{2}+\left(\frac{x\sqrt{3}}{2}\right)^{2}\right)+x^{2}=624$$

Ko te pūrua o \sqrt{3} ko 3.

$$4\left(16\times 3+8\sqrt{3}\times \frac{x\sqrt{3}}{2}+\left(\frac{x\sqrt{3}}{2}\right)^{2}\right)+x^{2}=624$$

Whakareatia te 16 ki te 3, ka 48.

$$4\left(48+8\sqrt{3}\times \frac{x\sqrt{3}}{2}+\left(\frac{x\sqrt{3}}{2}\right)^{2}\right)+x^{2}=624$$

Whakakorea atu te tauwehe pūnoa nui rawa 2 i roto i te 8 me te 2.

$$4\left(48+4x\sqrt{3}\sqrt{3}+\left(\frac{x\sqrt{3}}{2}\right)^{2}\right)+x^{2}=624$$

Kia whakarewa i te \frac{x\sqrt{3}}{2} ki tētahi taupū, me whakarewa tahi te taurunga me te tauraro ki te taupū kātahi ka whakawehe.

$$4\left(48+4x\sqrt{3}\sqrt{3}+\frac{\left(x\sqrt{3}\right)^{2}}{2^{2}}\right)+x^{2}=624$$

Hei tāpiri, hei tango kīanga rānei, me whakaroha ērā kia rite ā rātou tauraro. Whakareatia 48 ki te \frac{2^{2}}{2^{2}}.

$$4\left(\frac{48\times 2^{2}}{2^{2}}+4x\sqrt{3}\sqrt{3}+\frac{\left(x\sqrt{3}\right)^{2}}{2^{2}}\right)+x^{2}=624$$

Tā te mea he rite te tauraro o \frac{48\times 2^{2}}{2^{2}} me \frac{\left(x\sqrt{3}\right)^{2}}{2^{2}}, me tāpiri rāua mā te tāpiri i ō raua taurunga.

$$4\left(\frac{48\times 2^{2}+\left(x\sqrt{3}\right)^{2}}{2^{2}}+4x\sqrt{3}\sqrt{3}\right)+x^{2}=624$$

Whakamahia te āhuatanga tohatoha hei whakarea te 4 ki te \frac{48\times 2^{2}+\left(x\sqrt{3}\right)^{2}}{2^{2}}+4x\sqrt{3}\sqrt{3}.

$$4\times \frac{48\times 2^{2}+\left(x\sqrt{3}\right)^{2}}{2^{2}}+16\left(\sqrt{3}\right)^{2}x+x^{2}=624$$

Tātaihia te 2 mā te pū o 2, kia riro ko 4.

$$4\times \frac{48\times 4+\left(x\sqrt{3}\right)^{2}}{2^{2}}+16\left(\sqrt{3}\right)^{2}x+x^{2}=624$$

Whakareatia te 48 ki te 4, ka 192.

$$4\times \frac{192+\left(x\sqrt{3}\right)^{2}}{2^{2}}+16\left(\sqrt{3}\right)^{2}x+x^{2}=624$$

Whakarohaina te \left(x\sqrt{3}\right)^{2}.

$$4\times \frac{192+x^{2}\left(\sqrt{3}\right)^{2}}{2^{2}}+16\left(\sqrt{3}\right)^{2}x+x^{2}=624$$

Ko te pūrua o \sqrt{3} ko 3.

$$4\times \frac{192+x^{2}\times 3}{2^{2}}+16\left(\sqrt{3}\right)^{2}x+x^{2}=624$$

Tātaihia te 2 mā te pū o 2, kia riro ko 4.

$$4\times \frac{192+x^{2}\times 3}{4}+16\left(\sqrt{3}\right)^{2}x+x^{2}=624$$

Tuhia te 4\times \frac{192+x^{2}\times 3}{4} hei hautanga kotahi.

$$\frac{4\left(192+x^{2}\times 3\right)}{4}+16\left(\sqrt{3}\right)^{2}x+x^{2}=624$$

Me whakakore te 4 me te 4.

$$192+x^{2}\times 3+16\left(\sqrt{3}\right)^{2}x+x^{2}=624$$

Ko te pūrua o \sqrt{3} ko 3.

$$192+x^{2}\times 3+16\times 3x+x^{2}=624$$

Whakareatia te 16 ki te 3, ka 48.

$$192+x^{2}\times 3+48x+x^{2}=624$$

Pahekotia te x^{2}\times 3 me x^{2}, ka 4x^{2}.

$$192+4x^{2}+48x=624$$

Tangohia te 624 mai i ngā taha e rua.

$$192+4x^{2}+48x-624=0$$

Tangohia te 624 i te 192, ka -432.

$$-432+4x^{2}+48x=0$$

Whakawehea ngā taha e rua ki te 4.

$$-108+x^{2}+12x=0$$

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

$$x^{2}+12x-108=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx-108. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=12 ab=1\left(-108\right)=-108$$

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. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -108.

$$-1,108 -2,54 -3,36 -4,27 -6,18 -9,12$$

Tātaihia te tapeke mō ia takirua.

$$-1+108=107 -2+54=52 -3+36=33 -4+27=23 -6+18=12 -9+12=3$$

Ko te otinga te takirua ka hoatu i te tapeke 12.

$$a=-6 b=18$$

Tuhia anō te x^{2}+12x-108 hei \left(x^{2}-6x\right)+\left(18x-108\right).

$$\left(x^{2}-6x\right)+\left(18x-108\right)$$

Tauwehea te x i te tuatahi me te 18 i te rōpū tuarua.

$$x\left(x-6\right)+18\left(x-6\right)$$

Whakatauwehea atu te kīanga pātahi x-6 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-6\right)\left(x+18\right)$$

Hei kimi otinga whārite, me whakaoti te x-6=0 me te x+18=0.

$$x=6 x=-18$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakareatia ngā taha e rua o te whārite ki te 4.

$$4\left(4\sqrt{3}+\frac{x\sqrt{3}}{2}\right)^{2}+x^{2}=624$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(4\sqrt{3}+\frac{x\sqrt{3}}{2}\right)^{2}.

$$4\left(16\left(\sqrt{3}\right)^{2}+8\sqrt{3}\times \frac{x\sqrt{3}}{2}+\left(\frac{x\sqrt{3}}{2}\right)^{2}\right)+x^{2}=624$$

Ko te pūrua o \sqrt{3} ko 3.

$$4\left(16\times 3+8\sqrt{3}\times \frac{x\sqrt{3}}{2}+\left(\frac{x\sqrt{3}}{2}\right)^{2}\right)+x^{2}=624$$

Whakareatia te 16 ki te 3, ka 48.

$$4\left(48+8\sqrt{3}\times \frac{x\sqrt{3}}{2}+\left(\frac{x\sqrt{3}}{2}\right)^{2}\right)+x^{2}=624$$

Whakakorea atu te tauwehe pūnoa nui rawa 2 i roto i te 8 me te 2.

$$4\left(48+4x\sqrt{3}\sqrt{3}+\left(\frac{x\sqrt{3}}{2}\right)^{2}\right)+x^{2}=624$$

Kia whakarewa i te \frac{x\sqrt{3}}{2} ki tētahi taupū, me whakarewa tahi te taurunga me te tauraro ki te taupū kātahi ka whakawehe.

$$4\left(48+4x\sqrt{3}\sqrt{3}+\frac{\left(x\sqrt{3}\right)^{2}}{2^{2}}\right)+x^{2}=624$$

Hei tāpiri, hei tango kīanga rānei, me whakaroha ērā kia rite ā rātou tauraro. Whakareatia 48 ki te \frac{2^{2}}{2^{2}}.

$$4\left(\frac{48\times 2^{2}}{2^{2}}+4x\sqrt{3}\sqrt{3}+\frac{\left(x\sqrt{3}\right)^{2}}{2^{2}}\right)+x^{2}=624$$

Tā te mea he rite te tauraro o \frac{48\times 2^{2}}{2^{2}} me \frac{\left(x\sqrt{3}\right)^{2}}{2^{2}}, me tāpiri rāua mā te tāpiri i ō raua taurunga.

$$4\left(\frac{48\times 2^{2}+\left(x\sqrt{3}\right)^{2}}{2^{2}}+4x\sqrt{3}\sqrt{3}\right)+x^{2}=624$$

Whakamahia te āhuatanga tohatoha hei whakarea te 4 ki te \frac{48\times 2^{2}+\left(x\sqrt{3}\right)^{2}}{2^{2}}+4x\sqrt{3}\sqrt{3}.

$$4\times \frac{48\times 2^{2}+\left(x\sqrt{3}\right)^{2}}{2^{2}}+16\left(\sqrt{3}\right)^{2}x+x^{2}=624$$

Tātaihia te 2 mā te pū o 2, kia riro ko 4.

$$4\times \frac{48\times 4+\left(x\sqrt{3}\right)^{2}}{2^{2}}+16\left(\sqrt{3}\right)^{2}x+x^{2}=624$$

Whakareatia te 48 ki te 4, ka 192.

$$4\times \frac{192+\left(x\sqrt{3}\right)^{2}}{2^{2}}+16\left(\sqrt{3}\right)^{2}x+x^{2}=624$$

Whakarohaina te \left(x\sqrt{3}\right)^{2}.

$$4\times \frac{192+x^{2}\left(\sqrt{3}\right)^{2}}{2^{2}}+16\left(\sqrt{3}\right)^{2}x+x^{2}=624$$

Ko te pūrua o \sqrt{3} ko 3.

$$4\times \frac{192+x^{2}\times 3}{2^{2}}+16\left(\sqrt{3}\right)^{2}x+x^{2}=624$$

Tātaihia te 2 mā te pū o 2, kia riro ko 4.

$$4\times \frac{192+x^{2}\times 3}{4}+16\left(\sqrt{3}\right)^{2}x+x^{2}=624$$

Tuhia te 4\times \frac{192+x^{2}\times 3}{4} hei hautanga kotahi.

$$\frac{4\left(192+x^{2}\times 3\right)}{4}+16\left(\sqrt{3}\right)^{2}x+x^{2}=624$$

Me whakakore te 4 me te 4.

$$192+x^{2}\times 3+16\left(\sqrt{3}\right)^{2}x+x^{2}=624$$

Ko te pūrua o \sqrt{3} ko 3.

$$192+x^{2}\times 3+16\times 3x+x^{2}=624$$

Whakareatia te 16 ki te 3, ka 48.

$$192+x^{2}\times 3+48x+x^{2}=624$$

Pahekotia te x^{2}\times 3 me x^{2}, ka 4x^{2}.

$$192+4x^{2}+48x=624$$

Tangohia te 624 mai i ngā taha e rua.

$$192+4x^{2}+48x-624=0$$

Tangohia te 624 i te 192, ka -432.

$$-432+4x^{2}+48x=0$$

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.

$$4x^{2}+48x-432=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 4 mō a, 48 mō b, me -432 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-48±\sqrt{48^{2}-4\times 4\left(-432\right)}}{2\times 4}$$

Pūrua 48.

$$x=\frac{-48±\sqrt{2304-4\times 4\left(-432\right)}}{2\times 4}$$

Whakareatia -4 ki te 4.

$$x=\frac{-48±\sqrt{2304-16\left(-432\right)}}{2\times 4}$$

Whakareatia -16 ki te -432.

$$x=\frac{-48±\sqrt{2304+6912}}{2\times 4}$$

Tāpiri 2304 ki te 6912.

$$x=\frac{-48±\sqrt{9216}}{2\times 4}$$

Tuhia te pūtakerua o te 9216.

$$x=\frac{-48±96}{2\times 4}$$

Whakareatia 2 ki te 4.

$$x=\frac{-48±96}{8}$$

Nā, me whakaoti te whārite x=\frac{-48±96}{8} ina he tāpiri te ±. Tāpiri -48 ki te 96.

$$x=\frac{48}{8}$$

Whakawehe 48 ki te 8.

$$x=6$$

Nā, me whakaoti te whārite x=\frac{-48±96}{8} ina he tango te ±. Tango 96 mai i -48.

$$x=-\frac{144}{8}$$

Whakawehe -144 ki te 8.

$$x=-18$$

Kua oti te whārite te whakatau.

$$x=6 x=-18$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakareatia ngā taha e rua o te whārite ki te 4.

$$4\left(4\sqrt{3}+\frac{x\sqrt{3}}{2}\right)^{2}+x^{2}=624$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(4\sqrt{3}+\frac{x\sqrt{3}}{2}\right)^{2}.

$$4\left(16\left(\sqrt{3}\right)^{2}+8\sqrt{3}\times \frac{x\sqrt{3}}{2}+\left(\frac{x\sqrt{3}}{2}\right)^{2}\right)+x^{2}=624$$

Ko te pūrua o \sqrt{3} ko 3.

$$4\left(16\times 3+8\sqrt{3}\times \frac{x\sqrt{3}}{2}+\left(\frac{x\sqrt{3}}{2}\right)^{2}\right)+x^{2}=624$$

Whakareatia te 16 ki te 3, ka 48.

$$4\left(48+8\sqrt{3}\times \frac{x\sqrt{3}}{2}+\left(\frac{x\sqrt{3}}{2}\right)^{2}\right)+x^{2}=624$$

Whakakorea atu te tauwehe pūnoa nui rawa 2 i roto i te 8 me te 2.

$$4\left(48+4x\sqrt{3}\sqrt{3}+\left(\frac{x\sqrt{3}}{2}\right)^{2}\right)+x^{2}=624$$

Kia whakarewa i te \frac{x\sqrt{3}}{2} ki tētahi taupū, me whakarewa tahi te taurunga me te tauraro ki te taupū kātahi ka whakawehe.

$$4\left(48+4x\sqrt{3}\sqrt{3}+\frac{\left(x\sqrt{3}\right)^{2}}{2^{2}}\right)+x^{2}=624$$

Hei tāpiri, hei tango kīanga rānei, me whakaroha ērā kia rite ā rātou tauraro. Whakareatia 48 ki te \frac{2^{2}}{2^{2}}.

$$4\left(\frac{48\times 2^{2}}{2^{2}}+4x\sqrt{3}\sqrt{3}+\frac{\left(x\sqrt{3}\right)^{2}}{2^{2}}\right)+x^{2}=624$$

Tā te mea he rite te tauraro o \frac{48\times 2^{2}}{2^{2}} me \frac{\left(x\sqrt{3}\right)^{2}}{2^{2}}, me tāpiri rāua mā te tāpiri i ō raua taurunga.

$$4\left(\frac{48\times 2^{2}+\left(x\sqrt{3}\right)^{2}}{2^{2}}+4x\sqrt{3}\sqrt{3}\right)+x^{2}=624$$

Whakamahia te āhuatanga tohatoha hei whakarea te 4 ki te \frac{48\times 2^{2}+\left(x\sqrt{3}\right)^{2}}{2^{2}}+4x\sqrt{3}\sqrt{3}.

$$4\times \frac{48\times 2^{2}+\left(x\sqrt{3}\right)^{2}}{2^{2}}+16\left(\sqrt{3}\right)^{2}x+x^{2}=624$$

Tātaihia te 2 mā te pū o 2, kia riro ko 4.

$$4\times \frac{48\times 4+\left(x\sqrt{3}\right)^{2}}{2^{2}}+16\left(\sqrt{3}\right)^{2}x+x^{2}=624$$

Whakareatia te 48 ki te 4, ka 192.

$$4\times \frac{192+\left(x\sqrt{3}\right)^{2}}{2^{2}}+16\left(\sqrt{3}\right)^{2}x+x^{2}=624$$

Whakarohaina te \left(x\sqrt{3}\right)^{2}.

$$4\times \frac{192+x^{2}\left(\sqrt{3}\right)^{2}}{2^{2}}+16\left(\sqrt{3}\right)^{2}x+x^{2}=624$$

Ko te pūrua o \sqrt{3} ko 3.

$$4\times \frac{192+x^{2}\times 3}{2^{2}}+16\left(\sqrt{3}\right)^{2}x+x^{2}=624$$

Tātaihia te 2 mā te pū o 2, kia riro ko 4.

$$4\times \frac{192+x^{2}\times 3}{4}+16\left(\sqrt{3}\right)^{2}x+x^{2}=624$$

Tuhia te 4\times \frac{192+x^{2}\times 3}{4} hei hautanga kotahi.

$$\frac{4\left(192+x^{2}\times 3\right)}{4}+16\left(\sqrt{3}\right)^{2}x+x^{2}=624$$

Me whakakore te 4 me te 4.

$$192+x^{2}\times 3+16\left(\sqrt{3}\right)^{2}x+x^{2}=624$$

Ko te pūrua o \sqrt{3} ko 3.

$$192+x^{2}\times 3+16\times 3x+x^{2}=624$$

Whakareatia te 16 ki te 3, ka 48.

$$192+x^{2}\times 3+48x+x^{2}=624$$

Pahekotia te x^{2}\times 3 me x^{2}, ka 4x^{2}.

$$192+4x^{2}+48x=624$$

Tangohia te 192 mai i ngā taha e rua.

$$4x^{2}+48x=624-192$$

Tangohia te 192 i te 624, ka 432.

$$4x^{2}+48x=432$$

Whakawehea ngā taha e rua ki te 4.

$$\frac{4x^{2}+48x}{4}=\frac{432}{4}$$

Mā te whakawehe ki te 4 ka wetekia te whakareanga ki te 4.

$$x^{2}+\frac{48}{4}x=\frac{432}{4}$$

Whakawehe 48 ki te 4.

$$x^{2}+12x=\frac{432}{4}$$

Whakawehe 432 ki te 4.

$$x^{2}+12x=108$$

Whakawehea te 12, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 6. Nā, tāpiria te pūrua o te 6 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+12x+6^{2}=108+6^{2}$$

Pūrua 6.

$$x^{2}+12x+36=108+36$$

Tāpiri 108 ki te 36.

$$x^{2}+12x+36=144$$

Tauwehea x^{2}+12x+36. 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}.

$$\left(x+6\right)^{2}=144$$

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

$$\sqrt{\left(x+6\right)^{2}}=\sqrt{144}$$

Whakarūnātia.

$$x+6=12 x+6=-12$$

Me tango 6 mai i ngā taha e rua o te whārite.

$$x=6 x=-18$$