Whakaoti i te 2500=1600+left(6+x+xright)^2

$$2500=1600+ { \left(6+x+x \right) }^{ 2 }$$

Whakaoti mō x

$$x=12$$

$$x=-18$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Pahekotia te x me x, ka 2x.

$$2500=1600+\left(6+2x\right)^{2}$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(6+2x\right)^{2}.

$$2500=1600+36+24x+4x^{2}$$

Tāpirihia te 1600 ki te 36, ka 1636.

$$2500=1636+24x+4x^{2}$$

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$1636+24x+4x^{2}=2500$$

Tangohia te 2500 mai i ngā taha e rua.

$$1636+24x+4x^{2}-2500=0$$

Tangohia te 2500 i te 1636, ka -864.

$$-864+24x+4x^{2}=0$$

Whakawehea ngā taha e rua ki te 4.

$$-216+6x+x^{2}=0$$

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

$$x^{2}+6x-216=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx-216. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=6 ab=1\left(-216\right)=-216$$

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 -216.

$$-1,216 -2,108 -3,72 -4,54 -6,36 -8,27 -9,24 -12,18$$

Tātaihia te tapeke mō ia takirua.

$$-1+216=215 -2+108=106 -3+72=69 -4+54=50 -6+36=30 -8+27=19 -9+24=15 -12+18=6$$

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

$$a=-12 b=18$$

Tuhia anō te x^{2}+6x-216 hei \left(x^{2}-12x\right)+\left(18x-216\right).

$$\left(x^{2}-12x\right)+\left(18x-216\right)$$

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

$$x\left(x-12\right)+18\left(x-12\right)$$

Whakatauwehea atu te kīanga pātahi x-12 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-12\right)\left(x+18\right)$$

Hei kimi otinga whārite, me whakaoti te x-12=0 me te x+18=0.

$$x=12 x=-18$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Pahekotia te x me x, ka 2x.

$$2500=1600+\left(6+2x\right)^{2}$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(6+2x\right)^{2}.

$$2500=1600+36+24x+4x^{2}$$

Tāpirihia te 1600 ki te 36, ka 1636.

$$2500=1636+24x+4x^{2}$$

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$1636+24x+4x^{2}=2500$$

Tangohia te 2500 mai i ngā taha e rua.

$$1636+24x+4x^{2}-2500=0$$

Tangohia te 2500 i te 1636, ka -864.

$$-864+24x+4x^{2}=0$$

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.

$$4x^{2}+24x-864=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 4 mō a, 24 mō b, me -864 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-24±\sqrt{24^{2}-4\times 4\left(-864\right)}}{2\times 4}$$

Pūrua 24.

$$x=\frac{-24±\sqrt{576-4\times 4\left(-864\right)}}{2\times 4}$$

Whakareatia -4 ki te 4.

$$x=\frac{-24±\sqrt{576-16\left(-864\right)}}{2\times 4}$$

Whakareatia -16 ki te -864.

$$x=\frac{-24±\sqrt{576+13824}}{2\times 4}$$

Tāpiri 576 ki te 13824.

$$x=\frac{-24±\sqrt{14400}}{2\times 4}$$

Tuhia te pūtakerua o te 14400.

$$x=\frac{-24±120}{2\times 4}$$

Whakareatia 2 ki te 4.

$$x=\frac{-24±120}{8}$$

Nā, me whakaoti te whārite x=\frac{-24±120}{8} ina he tāpiri te ±. Tāpiri -24 ki te 120.

$$x=\frac{96}{8}$$

Whakawehe 96 ki te 8.

$$x=12$$

Nā, me whakaoti te whārite x=\frac{-24±120}{8} ina he tango te ±. Tango 120 mai i -24.

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

Whakawehe -144 ki te 8.

$$x=-18$$

Kua oti te whārite te whakatau.

$$x=12 x=-18$$

Ngā Upane mō te Whakaoti i te Pūrua

Pahekotia te x me x, ka 2x.

$$2500=1600+\left(6+2x\right)^{2}$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(6+2x\right)^{2}.

$$2500=1600+36+24x+4x^{2}$$

Tāpirihia te 1600 ki te 36, ka 1636.

$$2500=1636+24x+4x^{2}$$

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$1636+24x+4x^{2}=2500$$

Tangohia te 1636 mai i ngā taha e rua.

$$24x+4x^{2}=2500-1636$$

Tangohia te 1636 i te 2500, ka 864.

$$24x+4x^{2}=864$$

Ko ngā whārite pūrua pēnei i tēnei nā ka taea te whakaoti mā te whakaoti i te pūrua. Hei whakaoti i te pūrua, ko te whārite me mātua tuhi ki te āhua x^{2}+bx=c.

$$4x^{2}+24x=864$$

Whakawehea ngā taha e rua ki te 4.

$$\frac{4x^{2}+24x}{4}=\frac{864}{4}$$

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

$$x^{2}+\frac{24}{4}x=\frac{864}{4}$$

Whakawehe 24 ki te 4.

$$x^{2}+6x=\frac{864}{4}$$

Whakawehe 864 ki te 4.

$$x^{2}+6x=216$$

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.

$$x^{2}+6x+3^{2}=216+3^{2}$$

Pūrua 3.

$$x^{2}+6x+9=216+9$$

Tāpiri 216 ki te 9.

$$x^{2}+6x+9=225$$

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}.

$$\left(x+3\right)^{2}=225$$

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

$$\sqrt{\left(x+3\right)^{2}}=\sqrt{225}$$

Whakarūnātia.

$$x+3=15 x+3=-15$$

Me tango 3 mai i ngā taha e rua o te whārite.

$$x=12 x=-18$$