Whakaoti i te 5=1/2*20*x^2+1/2*50(x+0.2)^2

$$5 = \frac { 1 } { 2 } \cdot 20 \cdot x ^ { 2 } + \frac { 1 } { 2 } \cdot 50 ( x + 0.2 ) ^ { 2 }$$

Whakaoti mō x

$$x=\frac{\sqrt{165}}{35}-\frac{1}{7}\approx 0.224149502$$

$$x=-\frac{\sqrt{165}}{35}-\frac{1}{7}\approx -0.509863788$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakareatia te \frac{1}{2} ki te 20, ka 10.

$$5=10x^{2}+\frac{1}{2}\times 50\left(x+0.2\right)^{2}$$

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

$$5=10x^{2}+25\left(x+0.2\right)^{2}$$

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

$$5=10x^{2}+25\left(x^{2}+0.4x+0.04\right)$$

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

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

Pahekotia te 10x^{2} me 25x^{2}, ka 35x^{2}.

$$5=35x^{2}+10x+1$$

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

$$35x^{2}+10x+1=5$$

Tangohia te 5 mai i ngā taha e rua.

$$35x^{2}+10x+1-5=0$$

Tangohia te 5 i te 1, ka -4.

$$35x^{2}+10x-4=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 35 mō a, 10 mō b, me -4 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-10±\sqrt{10^{2}-4\times 35\left(-4\right)}}{2\times 35}$$

Pūrua 10.

$$x=\frac{-10±\sqrt{100-4\times 35\left(-4\right)}}{2\times 35}$$

Whakareatia -4 ki te 35.

$$x=\frac{-10±\sqrt{100-140\left(-4\right)}}{2\times 35}$$

Whakareatia -140 ki te -4.

$$x=\frac{-10±\sqrt{100+560}}{2\times 35}$$

Tāpiri 100 ki te 560.

$$x=\frac{-10±\sqrt{660}}{2\times 35}$$

Tuhia te pūtakerua o te 660.

$$x=\frac{-10±2\sqrt{165}}{2\times 35}$$

Whakareatia 2 ki te 35.

$$x=\frac{-10±2\sqrt{165}}{70}$$

Nā, me whakaoti te whārite x=\frac{-10±2\sqrt{165}}{70} ina he tāpiri te ±. Tāpiri -10 ki te 2\sqrt{165}.

$$x=\frac{2\sqrt{165}-10}{70}$$

Whakawehe -10+2\sqrt{165} ki te 70.

$$x=\frac{\sqrt{165}}{35}-\frac{1}{7}$$

Nā, me whakaoti te whārite x=\frac{-10±2\sqrt{165}}{70} ina he tango te ±. Tango 2\sqrt{165} mai i -10.

$$x=\frac{-2\sqrt{165}-10}{70}$$

Whakawehe -10-2\sqrt{165} ki te 70.

$$x=-\frac{\sqrt{165}}{35}-\frac{1}{7}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{165}}{35}-\frac{1}{7} x=-\frac{\sqrt{165}}{35}-\frac{1}{7}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakareatia te \frac{1}{2} ki te 20, ka 10.

$$5=10x^{2}+\frac{1}{2}\times 50\left(x+0.2\right)^{2}$$

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

$$5=10x^{2}+25\left(x+0.2\right)^{2}$$

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

$$5=10x^{2}+25\left(x^{2}+0.4x+0.04\right)$$

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

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

Pahekotia te 10x^{2} me 25x^{2}, ka 35x^{2}.

$$5=35x^{2}+10x+1$$

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

$$35x^{2}+10x+1=5$$

Tangohia te 1 mai i ngā taha e rua.

$$35x^{2}+10x=5-1$$

Tangohia te 1 i te 5, ka 4.

$$35x^{2}+10x=4$$

Whakawehea ngā taha e rua ki te 35.

$$\frac{35x^{2}+10x}{35}=\frac{4}{35}$$

Mā te whakawehe ki te 35 ka wetekia te whakareanga ki te 35.

$$x^{2}+\frac{10}{35}x=\frac{4}{35}$$

Whakahekea te hautanga \frac{10}{35} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 5.

$$x^{2}+\frac{2}{7}x=\frac{4}{35}$$

Whakawehea te \frac{2}{7}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{1}{7}. Nā, tāpiria te pūrua o te \frac{1}{7} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{2}{7}x+\left(\frac{1}{7}\right)^{2}=\frac{4}{35}+\left(\frac{1}{7}\right)^{2}$$

Pūruatia \frac{1}{7} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{2}{7}x+\frac{1}{49}=\frac{4}{35}+\frac{1}{49}$$

Tāpiri \frac{4}{35} ki te \frac{1}{49} 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.

$$x^{2}+\frac{2}{7}x+\frac{1}{49}=\frac{33}{245}$$

Tauwehea te x^{2}+\frac{2}{7}x+\frac{1}{49}. Ko te tikanga, ina ko x^{2}+bx+c he pūrua tika, ka taea te tauwehe i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{1}{7}\right)^{2}=\frac{33}{245}$$

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

$$\sqrt{\left(x+\frac{1}{7}\right)^{2}}=\sqrt{\frac{33}{245}}$$

Whakarūnātia.

$$x+\frac{1}{7}=\frac{\sqrt{165}}{35} x+\frac{1}{7}=-\frac{\sqrt{165}}{35}$$

Me tango \frac{1}{7} mai i ngā taha e rua o te whārite.

$$x=\frac{\sqrt{165}}{35}-\frac{1}{7} x=-\frac{\sqrt{165}}{35}-\frac{1}{7}$$