Whakaoti i te 200x^2+80x-9=0

$$200 x ^ { 2 } + 80 x - 9 = 0$$

Whakaoti mō x

$$x=\frac{\sqrt{34}}{20}-\frac{1}{5}\approx 0.091547595$$

$$x=-\frac{\sqrt{34}}{20}-\frac{1}{5}\approx -0.491547595$$

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.

$$200x^{2}+80x-9=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 200 mō a, 80 mō b, me -9 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-80±\sqrt{80^{2}-4\times 200\left(-9\right)}}{2\times 200}$$

Pūrua 80.

$$x=\frac{-80±\sqrt{6400-4\times 200\left(-9\right)}}{2\times 200}$$

Whakareatia -4 ki te 200.

$$x=\frac{-80±\sqrt{6400-800\left(-9\right)}}{2\times 200}$$

Whakareatia -800 ki te -9.

$$x=\frac{-80±\sqrt{6400+7200}}{2\times 200}$$

Tāpiri 6400 ki te 7200.

$$x=\frac{-80±\sqrt{13600}}{2\times 200}$$

Tuhia te pūtakerua o te 13600.

$$x=\frac{-80±20\sqrt{34}}{2\times 200}$$

Whakareatia 2 ki te 200.

$$x=\frac{-80±20\sqrt{34}}{400}$$

Nā, me whakaoti te whārite x=\frac{-80±20\sqrt{34}}{400} ina he tāpiri te ±. Tāpiri -80 ki te 20\sqrt{34}.

$$x=\frac{20\sqrt{34}-80}{400}$$

Whakawehe -80+20\sqrt{34} ki te 400.

$$x=\frac{\sqrt{34}}{20}-\frac{1}{5}$$

Nā, me whakaoti te whārite x=\frac{-80±20\sqrt{34}}{400} ina he tango te ±. Tango 20\sqrt{34} mai i -80.

$$x=\frac{-20\sqrt{34}-80}{400}$$

Whakawehe -80-20\sqrt{34} ki te 400.

$$x=-\frac{\sqrt{34}}{20}-\frac{1}{5}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{34}}{20}-\frac{1}{5} x=-\frac{\sqrt{34}}{20}-\frac{1}{5}$$

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.

$$200x^{2}+80x-9=0$$

Me tāpiri 9 ki ngā taha e rua o te whārite.

$$200x^{2}+80x-9-\left(-9\right)=-\left(-9\right)$$

Mā te tango i te -9 i a ia ake anō ka toe ko te 0.

$$200x^{2}+80x=-\left(-9\right)$$

Tango -9 mai i 0.

$$200x^{2}+80x=9$$

Whakawehea ngā taha e rua ki te 200.

$$\frac{200x^{2}+80x}{200}=\frac{9}{200}$$

Mā te whakawehe ki te 200 ka wetekia te whakareanga ki te 200.

$$x^{2}+\frac{80}{200}x=\frac{9}{200}$$

Whakahekea te hautanga \frac{80}{200} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 40.

$$x^{2}+\frac{2}{5}x=\frac{9}{200}$$

Whakawehea te \frac{2}{5}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{1}{5}. Nā, tāpiria te pūrua o te \frac{1}{5} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{2}{5}x+\left(\frac{1}{5}\right)^{2}=\frac{9}{200}+\left(\frac{1}{5}\right)^{2}$$

Pūruatia \frac{1}{5} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{2}{5}x+\frac{1}{25}=\frac{9}{200}+\frac{1}{25}$$

Tāpiri \frac{9}{200} ki te \frac{1}{25} 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.

$$x^{2}+\frac{2}{5}x+\frac{1}{25}=\frac{17}{200}$$

Tauwehea x^{2}+\frac{2}{5}x+\frac{1}{25}. 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}.

$$\left(x+\frac{1}{5}\right)^{2}=\frac{17}{200}$$

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

$$\sqrt{\left(x+\frac{1}{5}\right)^{2}}=\sqrt{\frac{17}{200}}$$

Whakarūnātia.

$$x+\frac{1}{5}=\frac{\sqrt{34}}{20} x+\frac{1}{5}=-\frac{\sqrt{34}}{20}$$

Me tango \frac{1}{5} mai i ngā taha e rua o te whārite.

$$x=\frac{\sqrt{34}}{20}-\frac{1}{5} x=-\frac{\sqrt{34}}{20}-\frac{1}{5}$$