Whakaoti i te 1/5x-3=5x1/10(x+1)

$$\frac { 1 } { 5 } x - 3 = 5 x \frac { 1 } { 10 } ( x + 1 )$$

Whakaoti mō x (complex solution)

$$x=\frac{-3+\sqrt{591}i}{10}\approx -0.3+2.431049156i$$

$$x=\frac{-\sqrt{591}i-3}{10}\approx -0.3-2.431049156i$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakareatia te 5 ki te \frac{1}{10}, ka \frac{5}{10}.

$$\frac{1}{5}x-3=\frac{5}{10}x\left(x+1\right)$$

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

$$\frac{1}{5}x-3=\frac{1}{2}x\left(x+1\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te \frac{1}{2}x ki te x+1.

$$\frac{1}{5}x-3=\frac{1}{2}xx+\frac{1}{2}x$$

Whakareatia te x ki te x, ka x^{2}.

$$\frac{1}{5}x-3=\frac{1}{2}x^{2}+\frac{1}{2}x$$

Tangohia te \frac{1}{2}x^{2} mai i ngā taha e rua.

$$\frac{1}{5}x-3-\frac{1}{2}x^{2}=\frac{1}{2}x$$

Tangohia te \frac{1}{2}x mai i ngā taha e rua.

$$\frac{1}{5}x-3-\frac{1}{2}x^{2}-\frac{1}{2}x=0$$

Pahekotia te \frac{1}{5}x me -\frac{1}{2}x, ka -\frac{3}{10}x.

$$-\frac{3}{10}x-3-\frac{1}{2}x^{2}=0$$

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.

$$-\frac{1}{2}x^{2}-\frac{3}{10}x-3=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -\frac{1}{2} mō a, -\frac{3}{10} mō b, me -3 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-\frac{3}{10}\right)±\sqrt{\left(-\frac{3}{10}\right)^{2}-4\left(-\frac{1}{2}\right)\left(-3\right)}}{2\left(-\frac{1}{2}\right)}$$

Pūruatia -\frac{3}{10} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x=\frac{-\left(-\frac{3}{10}\right)±\sqrt{\frac{9}{100}-4\left(-\frac{1}{2}\right)\left(-3\right)}}{2\left(-\frac{1}{2}\right)}$$

Whakareatia -4 ki te -\frac{1}{2}.

$$x=\frac{-\left(-\frac{3}{10}\right)±\sqrt{\frac{9}{100}+2\left(-3\right)}}{2\left(-\frac{1}{2}\right)}$$

Whakareatia 2 ki te -3.

$$x=\frac{-\left(-\frac{3}{10}\right)±\sqrt{\frac{9}{100}-6}}{2\left(-\frac{1}{2}\right)}$$

Tāpiri \frac{9}{100} ki te -6.

$$x=\frac{-\left(-\frac{3}{10}\right)±\sqrt{-\frac{591}{100}}}{2\left(-\frac{1}{2}\right)}$$

Tuhia te pūtakerua o te -\frac{591}{100}.

$$x=\frac{-\left(-\frac{3}{10}\right)±\frac{\sqrt{591}i}{10}}{2\left(-\frac{1}{2}\right)}$$

Ko te tauaro o -\frac{3}{10} ko \frac{3}{10}.

$$x=\frac{\frac{3}{10}±\frac{\sqrt{591}i}{10}}{2\left(-\frac{1}{2}\right)}$$

Whakareatia 2 ki te -\frac{1}{2}.

$$x=\frac{\frac{3}{10}±\frac{\sqrt{591}i}{10}}{-1}$$

Nā, me whakaoti te whārite x=\frac{\frac{3}{10}±\frac{\sqrt{591}i}{10}}{-1} ina he tāpiri te ±. Tāpiri \frac{3}{10} ki te \frac{i\sqrt{591}}{10}.

$$x=\frac{3+\sqrt{591}i}{-10}$$

Whakawehe \frac{3+i\sqrt{591}}{10} ki te -1.

$$x=\frac{-\sqrt{591}i-3}{10}$$

Nā, me whakaoti te whārite x=\frac{\frac{3}{10}±\frac{\sqrt{591}i}{10}}{-1} ina he tango te ±. Tango \frac{i\sqrt{591}}{10} mai i \frac{3}{10}.

$$x=\frac{-\sqrt{591}i+3}{-10}$$

Whakawehe \frac{3-i\sqrt{591}}{10} ki te -1.

$$x=\frac{-3+\sqrt{591}i}{10}$$

Kua oti te whārite te whakatau.

$$x=\frac{-\sqrt{591}i-3}{10} x=\frac{-3+\sqrt{591}i}{10}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakareatia te 5 ki te \frac{1}{10}, ka \frac{5}{10}.

$$\frac{1}{5}x-3=\frac{5}{10}x\left(x+1\right)$$

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

$$\frac{1}{5}x-3=\frac{1}{2}x\left(x+1\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te \frac{1}{2}x ki te x+1.

$$\frac{1}{5}x-3=\frac{1}{2}xx+\frac{1}{2}x$$

Whakareatia te x ki te x, ka x^{2}.

$$\frac{1}{5}x-3=\frac{1}{2}x^{2}+\frac{1}{2}x$$

Tangohia te \frac{1}{2}x^{2} mai i ngā taha e rua.

$$\frac{1}{5}x-3-\frac{1}{2}x^{2}=\frac{1}{2}x$$

Tangohia te \frac{1}{2}x mai i ngā taha e rua.

$$\frac{1}{5}x-3-\frac{1}{2}x^{2}-\frac{1}{2}x=0$$

Pahekotia te \frac{1}{5}x me -\frac{1}{2}x, ka -\frac{3}{10}x.

$$-\frac{3}{10}x-3-\frac{1}{2}x^{2}=0$$

Me tāpiri te 3 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$-\frac{3}{10}x-\frac{1}{2}x^{2}=3$$

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.

$$-\frac{1}{2}x^{2}-\frac{3}{10}x=3$$

Me whakarea ngā taha e rua ki te -2.

$$\frac{-\frac{1}{2}x^{2}-\frac{3}{10}x}{-\frac{1}{2}}=\frac{3}{-\frac{1}{2}}$$

Mā te whakawehe ki te -\frac{1}{2} ka wetekia te whakareanga ki te -\frac{1}{2}.

$$x^{2}+\left(-\frac{\frac{3}{10}}{-\frac{1}{2}}\right)x=\frac{3}{-\frac{1}{2}}$$

Whakawehe -\frac{3}{10} ki te -\frac{1}{2} mā te whakarea -\frac{3}{10} ki te tau huripoki o -\frac{1}{2}.

$$x^{2}+\frac{3}{5}x=\frac{3}{-\frac{1}{2}}$$

Whakawehe 3 ki te -\frac{1}{2} mā te whakarea 3 ki te tau huripoki o -\frac{1}{2}.

$$x^{2}+\frac{3}{5}x=-6$$

Whakawehea te \frac{3}{5}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{3}{10}. Nā, tāpiria te pūrua o te \frac{3}{10} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{3}{5}x+\left(\frac{3}{10}\right)^{2}=-6+\left(\frac{3}{10}\right)^{2}$$

Pūruatia \frac{3}{10} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{3}{5}x+\frac{9}{100}=-6+\frac{9}{100}$$

Tāpiri -6 ki te \frac{9}{100}.

$$x^{2}+\frac{3}{5}x+\frac{9}{100}=-\frac{591}{100}$$

Tauwehea x^{2}+\frac{3}{5}x+\frac{9}{100}. 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}.

$$\left(x+\frac{3}{10}\right)^{2}=-\frac{591}{100}$$

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

$$\sqrt{\left(x+\frac{3}{10}\right)^{2}}=\sqrt{-\frac{591}{100}}$$

Whakarūnātia.

$$x+\frac{3}{10}=\frac{\sqrt{591}i}{10} x+\frac{3}{10}=-\frac{\sqrt{591}i}{10}$$

Me tango \frac{3}{10} mai i ngā taha e rua o te whārite.

$$x=\frac{-3+\sqrt{591}i}{10} x=\frac{-\sqrt{591}i-3}{10}$$