Whakaoti i te -5x^2+9x=-3

$$- 5 x ^ { 2 } + 9 x = - 3$$

Whakaoti mō x

$$x = \frac{\sqrt{141} + 9}{10} \approx 2.087434209$$

$$x=\frac{9-\sqrt{141}}{10}\approx -0.287434209$$

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.

$$-5x^{2}+9x=-3$$

Me tāpiri 3 ki ngā taha e rua o te whārite.

$$-5x^{2}+9x-\left(-3\right)=-3-\left(-3\right)$$

Mā te tango i te -3 i a ia ake anō ka toe ko te 0.

$$-5x^{2}+9x-\left(-3\right)=0$$

Tango -3 mai i 0.

$$-5x^{2}+9x+3=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -5 mō a, 9 mō b, me 3 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-9±\sqrt{9^{2}-4\left(-5\right)\times 3}}{2\left(-5\right)}$$

Pūrua 9.

$$x=\frac{-9±\sqrt{81-4\left(-5\right)\times 3}}{2\left(-5\right)}$$

Whakareatia -4 ki te -5.

$$x=\frac{-9±\sqrt{81+20\times 3}}{2\left(-5\right)}$$

Whakareatia 20 ki te 3.

$$x=\frac{-9±\sqrt{81+60}}{2\left(-5\right)}$$

Tāpiri 81 ki te 60.

$$x=\frac{-9±\sqrt{141}}{2\left(-5\right)}$$

Whakareatia 2 ki te -5.

$$x=\frac{-9±\sqrt{141}}{-10}$$

Nā, me whakaoti te whārite x=\frac{-9±\sqrt{141}}{-10} ina he tāpiri te ±. Tāpiri -9 ki te \sqrt{141}.

$$x=\frac{\sqrt{141}-9}{-10}$$

Whakawehe -9+\sqrt{141} ki te -10.

$$x=\frac{9-\sqrt{141}}{10}$$

Nā, me whakaoti te whārite x=\frac{-9±\sqrt{141}}{-10} ina he tango te ±. Tango \sqrt{141} mai i -9.

$$x=\frac{-\sqrt{141}-9}{-10}$$

Whakawehe -9-\sqrt{141} ki te -10.

$$x=\frac{\sqrt{141}+9}{10}$$

Kua oti te whārite te whakatau.

$$x=\frac{9-\sqrt{141}}{10} x=\frac{\sqrt{141}+9}{10}$$

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.

$$-5x^{2}+9x=-3$$

Whakawehea ngā taha e rua ki te -5.

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

Mā te whakawehe ki te -5 ka wetekia te whakareanga ki te -5.

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

Whakawehe 9 ki te -5.

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

Whakawehe -3 ki te -5.

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

Whakawehea te -\frac{9}{5}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{9}{10}. Nā, tāpiria te pūrua o te -\frac{9}{10} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{9}{5}x+\left(-\frac{9}{10}\right)^{2}=\frac{3}{5}+\left(-\frac{9}{10}\right)^{2}$$

Pūruatia -\frac{9}{10} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{9}{5}x+\frac{81}{100}=\frac{3}{5}+\frac{81}{100}$$

Tāpiri \frac{3}{5} ki te \frac{81}{100} 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.

$$x^{2}-\frac{9}{5}x+\frac{81}{100}=\frac{141}{100}$$

Tauwehea x^{2}-\frac{9}{5}x+\frac{81}{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}.

$$\left(x-\frac{9}{10}\right)^{2}=\frac{141}{100}$$

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

$$\sqrt{\left(x-\frac{9}{10}\right)^{2}}=\sqrt{\frac{141}{100}}$$

Whakarūnātia.

$$x-\frac{9}{10}=\frac{\sqrt{141}}{10} x-\frac{9}{10}=-\frac{\sqrt{141}}{10}$$

Me tāpiri \frac{9}{10} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{141}+9}{10} x=\frac{9-\sqrt{141}}{10}$$