Whakaoti i te {left(05sqrt{3}xright)}^2+left(5-15xright)^2=left(1+xright)^2

$${ \left(05 \sqrt{ 3 } x \right) }^{ 2 } + { \left(5-15x \right) }^{ 2 } = { \left(1+x \right) }^{ 2 }$$

Whakaoti mō x

$$x=\frac{1}{4}=0.25$$

$$x=\frac{3}{7}\approx 0.428571429$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakareatia te 0 ki te 5, ka 0.

$$\left(0\sqrt{3}x\right)^{2}+\left(5-15x\right)^{2}=\left(1+x\right)^{2}$$

Ko te tau i whakarea ki te kore ka hua ko te kore.

$$0^{2}+\left(5-15x\right)^{2}=\left(1+x\right)^{2}$$

Tātaihia te 0 mā te pū o 2, kia riro ko 0.

$$0+\left(5-15x\right)^{2}=\left(1+x\right)^{2}$$

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(5-15x\right)^{2}.

$$0+25-150x+225x^{2}=\left(1+x\right)^{2}$$

Tāpirihia te 0 ki te 25, ka 25.

$$25-150x+225x^{2}=\left(1+x\right)^{2}$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(1+x\right)^{2}.

$$25-150x+225x^{2}=1+2x+x^{2}$$

Tangohia te 1 mai i ngā taha e rua.

$$25-150x+225x^{2}-1=2x+x^{2}$$

Tangohia te 1 i te 25, ka 24.

$$24-150x+225x^{2}=2x+x^{2}$$

Tangohia te 2x mai i ngā taha e rua.

$$24-150x+225x^{2}-2x=x^{2}$$

Pahekotia te -150x me -2x, ka -152x.

$$24-152x+225x^{2}=x^{2}$$

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

$$24-152x+225x^{2}-x^{2}=0$$

Pahekotia te 225x^{2} me -x^{2}, ka 224x^{2}.

$$24-152x+224x^{2}=0$$

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.

$$224x^{2}-152x+24=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 224 mō a, -152 mō b, me 24 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-152\right)±\sqrt{\left(-152\right)^{2}-4\times 224\times 24}}{2\times 224}$$

Pūrua -152.

$$x=\frac{-\left(-152\right)±\sqrt{23104-4\times 224\times 24}}{2\times 224}$$

Whakareatia -4 ki te 224.

$$x=\frac{-\left(-152\right)±\sqrt{23104-896\times 24}}{2\times 224}$$

Whakareatia -896 ki te 24.

$$x=\frac{-\left(-152\right)±\sqrt{23104-21504}}{2\times 224}$$

Tāpiri 23104 ki te -21504.

$$x=\frac{-\left(-152\right)±\sqrt{1600}}{2\times 224}$$

Tuhia te pūtakerua o te 1600.

$$x=\frac{-\left(-152\right)±40}{2\times 224}$$

Ko te tauaro o -152 ko 152.

$$x=\frac{152±40}{2\times 224}$$

Whakareatia 2 ki te 224.

$$x=\frac{152±40}{448}$$

Nā, me whakaoti te whārite x=\frac{152±40}{448} ina he tāpiri te ±. Tāpiri 152 ki te 40.

$$x=\frac{192}{448}$$

Whakahekea te hautanga \frac{192}{448} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 64.

$$x=\frac{3}{7}$$

Nā, me whakaoti te whārite x=\frac{152±40}{448} ina he tango te ±. Tango 40 mai i 152.

$$x=\frac{112}{448}$$

Whakahekea te hautanga \frac{112}{448} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 112.

$$x=\frac{1}{4}$$

Kua oti te whārite te whakatau.

$$x=\frac{3}{7} x=\frac{1}{4}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakareatia te 0 ki te 5, ka 0.

$$\left(0\sqrt{3}x\right)^{2}+\left(5-15x\right)^{2}=\left(1+x\right)^{2}$$

Ko te tau i whakarea ki te kore ka hua ko te kore.

$$0^{2}+\left(5-15x\right)^{2}=\left(1+x\right)^{2}$$

Tātaihia te 0 mā te pū o 2, kia riro ko 0.

$$0+\left(5-15x\right)^{2}=\left(1+x\right)^{2}$$

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(5-15x\right)^{2}.

$$0+25-150x+225x^{2}=\left(1+x\right)^{2}$$

Tāpirihia te 0 ki te 25, ka 25.

$$25-150x+225x^{2}=\left(1+x\right)^{2}$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(1+x\right)^{2}.

$$25-150x+225x^{2}=1+2x+x^{2}$$

Tangohia te 2x mai i ngā taha e rua.

$$25-150x+225x^{2}-2x=1+x^{2}$$

Pahekotia te -150x me -2x, ka -152x.

$$25-152x+225x^{2}=1+x^{2}$$

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

$$25-152x+225x^{2}-x^{2}=1$$

Pahekotia te 225x^{2} me -x^{2}, ka 224x^{2}.

$$25-152x+224x^{2}=1$$

Tangohia te 25 mai i ngā taha e rua.

$$-152x+224x^{2}=1-25$$

Tangohia te 25 i te 1, ka -24.

$$-152x+224x^{2}=-24$$

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.

$$224x^{2}-152x=-24$$

Whakawehea ngā taha e rua ki te 224.

$$\frac{224x^{2}-152x}{224}=-\frac{24}{224}$$

Mā te whakawehe ki te 224 ka wetekia te whakareanga ki te 224.

$$x^{2}+\left(-\frac{152}{224}\right)x=-\frac{24}{224}$$

Whakahekea te hautanga \frac{-152}{224} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 8.

$$x^{2}-\frac{19}{28}x=-\frac{24}{224}$$

Whakahekea te hautanga \frac{-24}{224} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 8.

$$x^{2}-\frac{19}{28}x=-\frac{3}{28}$$

Whakawehea te -\frac{19}{28}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{19}{56}. Nā, tāpiria te pūrua o te -\frac{19}{56} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{19}{28}x+\left(-\frac{19}{56}\right)^{2}=-\frac{3}{28}+\left(-\frac{19}{56}\right)^{2}$$

Pūruatia -\frac{19}{56} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{19}{28}x+\frac{361}{3136}=-\frac{3}{28}+\frac{361}{3136}$$

Tāpiri -\frac{3}{28} ki te \frac{361}{3136} 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.

$$x^{2}-\frac{19}{28}x+\frac{361}{3136}=\frac{25}{3136}$$

Tauwehea x^{2}-\frac{19}{28}x+\frac{361}{3136}. 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}.

$$\left(x-\frac{19}{56}\right)^{2}=\frac{25}{3136}$$

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

$$\sqrt{\left(x-\frac{19}{56}\right)^{2}}=\sqrt{\frac{25}{3136}}$$

Whakarūnātia.

$$x-\frac{19}{56}=\frac{5}{56} x-\frac{19}{56}=-\frac{5}{56}$$

Me tāpiri \frac{19}{56} ki ngā taha e rua o te whārite.

$$x=\frac{3}{7} x=\frac{1}{4}$$