Whakaoti i te 5x^2-13x-30=0

$$5 x ^ { 2 } - 13 x - 30 = 0$$

Whakaoti mō x

$$x = \frac{\sqrt{769} + 13}{10} \approx 4.073084925$$

$$x=\frac{13-\sqrt{769}}{10}\approx -1.473084925$$

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.

$$5x^{2}-13x-30=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 5 mō a, -13 mō b, me -30 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-13\right)±\sqrt{\left(-13\right)^{2}-4\times 5\left(-30\right)}}{2\times 5}$$

Pūrua -13.

$$x=\frac{-\left(-13\right)±\sqrt{169-4\times 5\left(-30\right)}}{2\times 5}$$

Whakareatia -4 ki te 5.

$$x=\frac{-\left(-13\right)±\sqrt{169-20\left(-30\right)}}{2\times 5}$$

Whakareatia -20 ki te -30.

$$x=\frac{-\left(-13\right)±\sqrt{169+600}}{2\times 5}$$

Tāpiri 169 ki te 600.

$$x=\frac{-\left(-13\right)±\sqrt{769}}{2\times 5}$$

Ko te tauaro o -13 ko 13.

$$x=\frac{13±\sqrt{769}}{2\times 5}$$

Whakareatia 2 ki te 5.

$$x=\frac{13±\sqrt{769}}{10}$$

Nā, me whakaoti te whārite x=\frac{13±\sqrt{769}}{10} ina he tāpiri te ±. Tāpiri 13 ki te \sqrt{769}.

$$x=\frac{\sqrt{769}+13}{10}$$

Nā, me whakaoti te whārite x=\frac{13±\sqrt{769}}{10} ina he tango te ±. Tango \sqrt{769} mai i 13.

$$x=\frac{13-\sqrt{769}}{10}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{769}+13}{10} x=\frac{13-\sqrt{769}}{10}$$

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.

$$5x^{2}-13x-30=0$$

Me tāpiri 30 ki ngā taha e rua o te whārite.

$$5x^{2}-13x-30-\left(-30\right)=-\left(-30\right)$$

Mā te tango i te -30 i a ia ake anō ka toe ko te 0.

$$5x^{2}-13x=-\left(-30\right)$$

Tango -30 mai i 0.

$$5x^{2}-13x=30$$

Whakawehea ngā taha e rua ki te 5.

$$\frac{5x^{2}-13x}{5}=\frac{30}{5}$$

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

$$x^{2}-\frac{13}{5}x=\frac{30}{5}$$

Whakawehe 30 ki te 5.

$$x^{2}-\frac{13}{5}x=6$$

Whakawehea te -\frac{13}{5}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{13}{10}. Nā, tāpiria te pūrua o te -\frac{13}{10} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{13}{5}x+\left(-\frac{13}{10}\right)^{2}=6+\left(-\frac{13}{10}\right)^{2}$$

Pūruatia -\frac{13}{10} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{13}{5}x+\frac{169}{100}=6+\frac{169}{100}$$

Tāpiri 6 ki te \frac{169}{100}.

$$x^{2}-\frac{13}{5}x+\frac{169}{100}=\frac{769}{100}$$

Tauwehea x^{2}-\frac{13}{5}x+\frac{169}{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}.

$$\left(x-\frac{13}{10}\right)^{2}=\frac{769}{100}$$

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

$$\sqrt{\left(x-\frac{13}{10}\right)^{2}}=\sqrt{\frac{769}{100}}$$

Whakarūnātia.

$$x-\frac{13}{10}=\frac{\sqrt{769}}{10} x-\frac{13}{10}=-\frac{\sqrt{769}}{10}$$

Me tāpiri \frac{13}{10} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{769}+13}{10} x=\frac{13-\sqrt{769}}{10}$$