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

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

Whakaoti mō x

$$x = \frac{3 \sqrt{21} + 3}{10} \approx 1.674772708$$

$$x=\frac{3-3\sqrt{21}}{10}\approx -1.074772708$$

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.

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

Me tango 9 mai i ngā taha e rua o te whārite.

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

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

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

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

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

Pūrua -3.

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

Whakareatia -4 ki te 5.

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

Whakareatia -20 ki te -9.

$$x=\frac{-\left(-3\right)±\sqrt{9+180}}{2\times 5}$$

Tāpiri 9 ki te 180.

$$x=\frac{-\left(-3\right)±\sqrt{189}}{2\times 5}$$

Tuhia te pūtakerua o te 189.

$$x=\frac{-\left(-3\right)±3\sqrt{21}}{2\times 5}$$

Ko te tauaro o -3 ko 3.

$$x=\frac{3±3\sqrt{21}}{2\times 5}$$

Whakareatia 2 ki te 5.

$$x=\frac{3±3\sqrt{21}}{10}$$

Nā, me whakaoti te whārite x=\frac{3±3\sqrt{21}}{10} ina he tāpiri te ±. Tāpiri 3 ki te 3\sqrt{21}.

$$x=\frac{3\sqrt{21}+3}{10}$$

Nā, me whakaoti te whārite x=\frac{3±3\sqrt{21}}{10} ina he tango te ±. Tango 3\sqrt{21} mai i 3.

$$x=\frac{3-3\sqrt{21}}{10}$$

Kua oti te whārite te whakatau.

$$x=\frac{3\sqrt{21}+3}{10} x=\frac{3-3\sqrt{21}}{10}$$

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.

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

Whakawehea ngā taha e rua ki te 5.

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

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

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

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.

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

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

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

Tāpiri \frac{9}{5} ki te \frac{9}{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.

$$x^{2}-\frac{3}{5}x+\frac{9}{100}=\frac{189}{100}$$

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}.

$$\left(x-\frac{3}{10}\right)^{2}=\frac{189}{100}$$

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

$$\sqrt{\left(x-\frac{3}{10}\right)^{2}}=\sqrt{\frac{189}{100}}$$

Whakarūnātia.

$$x-\frac{3}{10}=\frac{3\sqrt{21}}{10} x-\frac{3}{10}=-\frac{3\sqrt{21}}{10}$$

Me tāpiri \frac{3}{10} ki ngā taha e rua o te whārite.

$$x=\frac{3\sqrt{21}+3}{10} x=\frac{3-3\sqrt{21}}{10}$$