Whakaoti i te 5=1/2250x^2+1/250left(x+0.2right)^2

$$5 = \frac{ 1 }{ 2 } 250 { x }^{ 2 } + \frac{ 1 }{ 2 } 50 { \left(x+0.2 \right) }^{ 2 }$$

Whakaoti mō x

$$x=\frac{2}{15}\approx 0.133333333$$

$$x=-0.2$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakareatia te \frac{1}{2} ki te 250, ka 125.

$$5=125x^{2}+\frac{1}{2}\times 50\left(x+0.2\right)^{2}$$

Whakareatia te \frac{1}{2} ki te 50, ka 25.

$$5=125x^{2}+25\left(x+0.2\right)^{2}$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(x+0.2\right)^{2}.

$$5=125x^{2}+25\left(x^{2}+0.4x+0.04\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 25 ki te x^{2}+0.4x+0.04.

$$5=125x^{2}+25x^{2}+10x+1$$

Pahekotia te 125x^{2} me 25x^{2}, ka 150x^{2}.

$$5=150x^{2}+10x+1$$

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

$$150x^{2}+10x+1=5$$

Tangohia te 5 mai i ngā taha e rua.

$$150x^{2}+10x+1-5=0$$

Tangohia te 5 i te 1, ka -4.

$$150x^{2}+10x-4=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 150x^{2}+ax+bx-4. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=10 ab=150\left(-4\right)=-600$$

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 -600.

$$-1,600 -2,300 -3,200 -4,150 -5,120 -6,100 -8,75 -10,60 -12,50 -15,40 -20,30 -24,25$$

Tātaihia te tapeke mō ia takirua.

$$-1+600=599 -2+300=298 -3+200=197 -4+150=146 -5+120=115 -6+100=94 -8+75=67 -10+60=50 -12+50=38 -15+40=25 -20+30=10 -24+25=1$$

Ko te otinga te takirua ka hoatu i te tapeke 5.

$$a=-10 b=15$$

Tuhia anō te 150x^{2}+10x-4 hei \left(150x^{2}-10x\right)+\left(15x-4\right).

$$\left(150x^{2}-10x\right)+\left(15x-4\right)$$

Whakatauwehea atu 5x i te 150x^{2}-10x.

$$5x\left(15x-2\right)+15x-2$$

Whakatauwehea atu te kīanga pātahi 15x-2 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(15x-2\right)\left(5x+1\right)$$

Hei kimi otinga whārite, me whakaoti te 15x-2=0 me te 5x+1=0.

$$x=\frac{2}{15} x=-\frac{1}{5}$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakareatia te \frac{1}{2} ki te 250, ka 125.

$$5=125x^{2}+\frac{1}{2}\times 50\left(x+0.2\right)^{2}$$

Whakareatia te \frac{1}{2} ki te 50, ka 25.

$$5=125x^{2}+25\left(x+0.2\right)^{2}$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(x+0.2\right)^{2}.

$$5=125x^{2}+25\left(x^{2}+0.4x+0.04\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 25 ki te x^{2}+0.4x+0.04.

$$5=125x^{2}+25x^{2}+10x+1$$

Pahekotia te 125x^{2} me 25x^{2}, ka 150x^{2}.

$$5=150x^{2}+10x+1$$

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

$$150x^{2}+10x+1=5$$

Tangohia te 5 mai i ngā taha e rua.

$$150x^{2}+10x+1-5=0$$

Tangohia te 5 i te 1, ka -4.

$$150x^{2}+10x-4=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 150 mō a, 10 mō b, me -4 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-10±\sqrt{10^{2}-4\times 150\left(-4\right)}}{2\times 150}$$

Pūrua 10.

$$x=\frac{-10±\sqrt{100-4\times 150\left(-4\right)}}{2\times 150}$$

Whakareatia -4 ki te 150.

$$x=\frac{-10±\sqrt{100-600\left(-4\right)}}{2\times 150}$$

Whakareatia -600 ki te -4.

$$x=\frac{-10±\sqrt{100+2400}}{2\times 150}$$

Tāpiri 100 ki te 2400.

$$x=\frac{-10±\sqrt{2500}}{2\times 150}$$

Tuhia te pūtakerua o te 2500.

$$x=\frac{-10±50}{2\times 150}$$

Whakareatia 2 ki te 150.

$$x=\frac{-10±50}{300}$$

Nā, me whakaoti te whārite x=\frac{-10±50}{300} ina he tāpiri te ±. Tāpiri -10 ki te 50.

$$x=\frac{40}{300}$$

Whakahekea te hautanga \frac{40}{300} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 20.

$$x=\frac{2}{15}$$

Nā, me whakaoti te whārite x=\frac{-10±50}{300} ina he tango te ±. Tango 50 mai i -10.

$$x=-\frac{60}{300}$$

Whakahekea te hautanga \frac{-60}{300} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 60.

$$x=-\frac{1}{5}$$

Kua oti te whārite te whakatau.

$$x=\frac{2}{15} x=-\frac{1}{5}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakareatia te \frac{1}{2} ki te 250, ka 125.

$$5=125x^{2}+\frac{1}{2}\times 50\left(x+0.2\right)^{2}$$

Whakareatia te \frac{1}{2} ki te 50, ka 25.

$$5=125x^{2}+25\left(x+0.2\right)^{2}$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(x+0.2\right)^{2}.

$$5=125x^{2}+25\left(x^{2}+0.4x+0.04\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 25 ki te x^{2}+0.4x+0.04.

$$5=125x^{2}+25x^{2}+10x+1$$

Pahekotia te 125x^{2} me 25x^{2}, ka 150x^{2}.

$$5=150x^{2}+10x+1$$

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

$$150x^{2}+10x+1=5$$

Tangohia te 1 mai i ngā taha e rua.

$$150x^{2}+10x=5-1$$

Tangohia te 1 i te 5, ka 4.

$$150x^{2}+10x=4$$

Whakawehea ngā taha e rua ki te 150.

$$\frac{150x^{2}+10x}{150}=\frac{4}{150}$$

Mā te whakawehe ki te 150 ka wetekia te whakareanga ki te 150.

$$x^{2}+\frac{10}{150}x=\frac{4}{150}$$

Whakahekea te hautanga \frac{10}{150} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 10.

$$x^{2}+\frac{1}{15}x=\frac{4}{150}$$

Whakahekea te hautanga \frac{4}{150} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x^{2}+\frac{1}{15}x=\frac{2}{75}$$

Whakawehea te \frac{1}{15}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{1}{30}. Nā, tāpiria te pūrua o te \frac{1}{30} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{1}{15}x+\left(\frac{1}{30}\right)^{2}=\frac{2}{75}+\left(\frac{1}{30}\right)^{2}$$

Pūruatia \frac{1}{30} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{1}{15}x+\frac{1}{900}=\frac{2}{75}+\frac{1}{900}$$

Tāpiri \frac{2}{75} ki te \frac{1}{900} 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.

$$x^{2}+\frac{1}{15}x+\frac{1}{900}=\frac{1}{36}$$

Tauwehea x^{2}+\frac{1}{15}x+\frac{1}{900}. 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}.

$$\left(x+\frac{1}{30}\right)^{2}=\frac{1}{36}$$

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

$$\sqrt{\left(x+\frac{1}{30}\right)^{2}}=\sqrt{\frac{1}{36}}$$

Whakarūnātia.

$$x+\frac{1}{30}=\frac{1}{6} x+\frac{1}{30}=-\frac{1}{6}$$

Me tango \frac{1}{30} mai i ngā taha e rua o te whārite.

$$x=\frac{2}{15} x=-\frac{1}{5}$$