Whakaoti i te (100)^2+(x+100)^2=(2x+100)^2

$$( 100 ) ^ { 2 } + ( x + 100 ) ^ { 2 } = ( 2 x + 100 ) ^ { 2 }$$

Whakaoti mō x

$$x = \frac{100}{3} = 33\frac{1}{3} \approx 33.333333333$$

$$x=-100$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tātaihia te 100 mā te pū o 2, kia riro ko 10000.

$$10000+\left(x+100\right)^{2}=\left(2x+100\right)^{2}$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(x+100\right)^{2}.

$$10000+x^{2}+200x+10000=\left(2x+100\right)^{2}$$

Tāpirihia te 10000 ki te 10000, ka 20000.

$$20000+x^{2}+200x=\left(2x+100\right)^{2}$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(2x+100\right)^{2}.

$$20000+x^{2}+200x=4x^{2}+400x+10000$$

Tangohia te 4x^{2} mai i ngā taha e rua.

$$20000+x^{2}+200x-4x^{2}=400x+10000$$

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

$$20000-3x^{2}+200x=400x+10000$$

Tangohia te 400x mai i ngā taha e rua.

$$20000-3x^{2}+200x-400x=10000$$

Pahekotia te 200x me -400x, ka -200x.

$$20000-3x^{2}-200x=10000$$

Tangohia te 10000 mai i ngā taha e rua.

$$20000-3x^{2}-200x-10000=0$$

Tangohia te 10000 i te 20000, ka 10000.

$$10000-3x^{2}-200x=0$$

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

$$-3x^{2}-200x+10000=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei -3x^{2}+ax+bx+10000. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-200 ab=-3\times 10000=-30000$$

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 -30000.

$$1,-30000 2,-15000 3,-10000 4,-7500 5,-6000 6,-5000 8,-3750 10,-3000 12,-2500 15,-2000 16,-1875 20,-1500 24,-1250 25,-1200 30,-1000 40,-750 48,-625 50,-600 60,-500 75,-400 80,-375 100,-300 120,-250 125,-240 150,-200$$

Tātaihia te tapeke mō ia takirua.

$$1-30000=-29999 2-15000=-14998 3-10000=-9997 4-7500=-7496 5-6000=-5995 6-5000=-4994 8-3750=-3742 10-3000=-2990 12-2500=-2488 15-2000=-1985 16-1875=-1859 20-1500=-1480 24-1250=-1226 25-1200=-1175 30-1000=-970 40-750=-710 48-625=-577 50-600=-550 60-500=-440 75-400=-325 80-375=-295 100-300=-200 120-250=-130 125-240=-115 150-200=-50$$

Ko te otinga te takirua ka hoatu i te tapeke -200.

$$a=100 b=-300$$

Tuhia anō te -3x^{2}-200x+10000 hei \left(-3x^{2}+100x\right)+\left(-300x+10000\right).

$$\left(-3x^{2}+100x\right)+\left(-300x+10000\right)$$

Tauwehea te -x i te tuatahi me te -100 i te rōpū tuarua.

$$-x\left(3x-100\right)-100\left(3x-100\right)$$

Whakatauwehea atu te kīanga pātahi 3x-100 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(3x-100\right)\left(-x-100\right)$$

Hei kimi otinga whārite, me whakaoti te 3x-100=0 me te -x-100=0.

$$x=\frac{100}{3} x=-100$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tātaihia te 100 mā te pū o 2, kia riro ko 10000.

$$10000+\left(x+100\right)^{2}=\left(2x+100\right)^{2}$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(x+100\right)^{2}.

$$10000+x^{2}+200x+10000=\left(2x+100\right)^{2}$$

Tāpirihia te 10000 ki te 10000, ka 20000.

$$20000+x^{2}+200x=\left(2x+100\right)^{2}$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(2x+100\right)^{2}.

$$20000+x^{2}+200x=4x^{2}+400x+10000$$

Tangohia te 4x^{2} mai i ngā taha e rua.

$$20000+x^{2}+200x-4x^{2}=400x+10000$$

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

$$20000-3x^{2}+200x=400x+10000$$

Tangohia te 400x mai i ngā taha e rua.

$$20000-3x^{2}+200x-400x=10000$$

Pahekotia te 200x me -400x, ka -200x.

$$20000-3x^{2}-200x=10000$$

Tangohia te 10000 mai i ngā taha e rua.

$$20000-3x^{2}-200x-10000=0$$

Tangohia te 10000 i te 20000, ka 10000.

$$10000-3x^{2}-200x=0$$

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.

$$-3x^{2}-200x+10000=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -3 mō a, -200 mō b, me 10000 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-200\right)±\sqrt{\left(-200\right)^{2}-4\left(-3\right)\times 10000}}{2\left(-3\right)}$$

Pūrua -200.

$$x=\frac{-\left(-200\right)±\sqrt{40000-4\left(-3\right)\times 10000}}{2\left(-3\right)}$$

Whakareatia -4 ki te -3.

$$x=\frac{-\left(-200\right)±\sqrt{40000+12\times 10000}}{2\left(-3\right)}$$

Whakareatia 12 ki te 10000.

$$x=\frac{-\left(-200\right)±\sqrt{40000+120000}}{2\left(-3\right)}$$

Tāpiri 40000 ki te 120000.

$$x=\frac{-\left(-200\right)±\sqrt{160000}}{2\left(-3\right)}$$

Tuhia te pūtakerua o te 160000.

$$x=\frac{-\left(-200\right)±400}{2\left(-3\right)}$$

Ko te tauaro o -200 ko 200.

$$x=\frac{200±400}{2\left(-3\right)}$$

Whakareatia 2 ki te -3.

$$x=\frac{200±400}{-6}$$

Nā, me whakaoti te whārite x=\frac{200±400}{-6} ina he tāpiri te ±. Tāpiri 200 ki te 400.

$$x=\frac{600}{-6}$$

Whakawehe 600 ki te -6.

$$x=-100$$

Nā, me whakaoti te whārite x=\frac{200±400}{-6} ina he tango te ±. Tango 400 mai i 200.

$$x=-\frac{200}{-6}$$

Whakahekea te hautanga \frac{-200}{-6} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x=\frac{100}{3}$$

Kua oti te whārite te whakatau.

$$x=-100 x=\frac{100}{3}$$

Ngā Upane mō te Whakaoti i te Pūrua

Tātaihia te 100 mā te pū o 2, kia riro ko 10000.

$$10000+\left(x+100\right)^{2}=\left(2x+100\right)^{2}$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(x+100\right)^{2}.

$$10000+x^{2}+200x+10000=\left(2x+100\right)^{2}$$

Tāpirihia te 10000 ki te 10000, ka 20000.

$$20000+x^{2}+200x=\left(2x+100\right)^{2}$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(2x+100\right)^{2}.

$$20000+x^{2}+200x=4x^{2}+400x+10000$$

Tangohia te 4x^{2} mai i ngā taha e rua.

$$20000+x^{2}+200x-4x^{2}=400x+10000$$

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

$$20000-3x^{2}+200x=400x+10000$$

Tangohia te 400x mai i ngā taha e rua.

$$20000-3x^{2}+200x-400x=10000$$

Pahekotia te 200x me -400x, ka -200x.

$$20000-3x^{2}-200x=10000$$

Tangohia te 20000 mai i ngā taha e rua.

$$-3x^{2}-200x=10000-20000$$

Tangohia te 20000 i te 10000, ka -10000.

$$-3x^{2}-200x=-10000$$

Whakawehea ngā taha e rua ki te -3.

$$\frac{-3x^{2}-200x}{-3}=-\frac{10000}{-3}$$

Mā te whakawehe ki te -3 ka wetekia te whakareanga ki te -3.

$$x^{2}+\left(-\frac{200}{-3}\right)x=-\frac{10000}{-3}$$

Whakawehe -200 ki te -3.

$$x^{2}+\frac{200}{3}x=-\frac{10000}{-3}$$

Whakawehe -10000 ki te -3.

$$x^{2}+\frac{200}{3}x=\frac{10000}{3}$$

Whakawehea te \frac{200}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{100}{3}. Nā, tāpiria te pūrua o te \frac{100}{3} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{200}{3}x+\left(\frac{100}{3}\right)^{2}=\frac{10000}{3}+\left(\frac{100}{3}\right)^{2}$$

Pūruatia \frac{100}{3} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{200}{3}x+\frac{10000}{9}=\frac{10000}{3}+\frac{10000}{9}$$

Tāpiri \frac{10000}{3} ki te \frac{10000}{9} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x^{2}+\frac{200}{3}x+\frac{10000}{9}=\frac{40000}{9}$$

Tauwehea x^{2}+\frac{200}{3}x+\frac{10000}{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}.

$$\left(x+\frac{100}{3}\right)^{2}=\frac{40000}{9}$$

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

$$\sqrt{\left(x+\frac{100}{3}\right)^{2}}=\sqrt{\frac{40000}{9}}$$

Whakarūnātia.

$$x+\frac{100}{3}=\frac{200}{3} x+\frac{100}{3}=-\frac{200}{3}$$

Me tango \frac{100}{3} mai i ngā taha e rua o te whārite.

$$x=\frac{100}{3} x=-100$$