Whakaoti i te x+x^2=5/109=

$$x + x ^ { 2 } = \frac { 5 } { 109 } =$$

Whakaoti mō x

$$x=\frac{\sqrt{14061}}{218}-\frac{1}{2}\approx 0.043940768$$

$$x=-\frac{\sqrt{14061}}{218}-\frac{1}{2}\approx -1.043940768$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

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.

$$x^{2}+x=\frac{5}{109}$$

Me tango \frac{5}{109} mai i ngā taha e rua o te whārite.

$$x^{2}+x-\frac{5}{109}=\frac{5}{109}-\frac{5}{109}$$

Mā te tango i te \frac{5}{109} i a ia ake anō ka toe ko te 0.

$$x^{2}+x-\frac{5}{109}=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 1 mō b, me -\frac{5}{109} mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1±\sqrt{1^{2}-4\left(-\frac{5}{109}\right)}}{2}$$

Pūrua 1.

$$x=\frac{-1±\sqrt{1-4\left(-\frac{5}{109}\right)}}{2}$$

Whakareatia -4 ki te -\frac{5}{109}.

$$x=\frac{-1±\sqrt{1+\frac{20}{109}}}{2}$$

Tāpiri 1 ki te \frac{20}{109}.

$$x=\frac{-1±\sqrt{\frac{129}{109}}}{2}$$

Tuhia te pūtakerua o te \frac{129}{109}.

$$x=\frac{-1±\frac{\sqrt{14061}}{109}}{2}$$

Nā, me whakaoti te whārite x=\frac{-1±\frac{\sqrt{14061}}{109}}{2} ina he tāpiri te ±. Tāpiri -1 ki te \frac{\sqrt{14061}}{109}.

$$x=\frac{\frac{\sqrt{14061}}{109}-1}{2}$$

Whakawehe -1+\frac{\sqrt{14061}}{109} ki te 2.

$$x=\frac{\sqrt{14061}}{218}-\frac{1}{2}$$

Nā, me whakaoti te whārite x=\frac{-1±\frac{\sqrt{14061}}{109}}{2} ina he tango te ±. Tango \frac{\sqrt{14061}}{109} mai i -1.

$$x=\frac{-\frac{\sqrt{14061}}{109}-1}{2}$$

Whakawehe -1-\frac{\sqrt{14061}}{109} ki te 2.

$$x=-\frac{\sqrt{14061}}{218}-\frac{1}{2}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{14061}}{218}-\frac{1}{2} x=-\frac{\sqrt{14061}}{218}-\frac{1}{2}$$

Ngā Upane mō te Whakaoti i te Pūrua

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.

$$x^{2}+x=\frac{5}{109}$$

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

$$x^{2}+x+\left(\frac{1}{2}\right)^{2}=\frac{5}{109}+\left(\frac{1}{2}\right)^{2}$$

Pūruatia \frac{1}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+x+\frac{1}{4}=\frac{5}{109}+\frac{1}{4}$$

Tāpiri \frac{5}{109} ki te \frac{1}{4} 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.

$$x^{2}+x+\frac{1}{4}=\frac{129}{436}$$

Tauwehea x^{2}+x+\frac{1}{4}. 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}.

$$\left(x+\frac{1}{2}\right)^{2}=\frac{129}{436}$$

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

$$\sqrt{\left(x+\frac{1}{2}\right)^{2}}=\sqrt{\frac{129}{436}}$$

Whakarūnātia.

$$x+\frac{1}{2}=\frac{\sqrt{14061}}{218} x+\frac{1}{2}=-\frac{\sqrt{14061}}{218}$$

Me tango \frac{1}{2} mai i ngā taha e rua o te whārite.

$$x=\frac{\sqrt{14061}}{218}-\frac{1}{2} x=-\frac{\sqrt{14061}}{218}-\frac{1}{2}$$