Whakaoti i te 5x^2-5x-17=0

$$5 x ^ { 2 } - 5 x - 17 = 0$$

Whakaoti mō x

$$x=\frac{\sqrt{365}}{10}+\frac{1}{2}\approx 2.410497317$$

$$x=-\frac{\sqrt{365}}{10}+\frac{1}{2}\approx -1.410497317$$

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.

$$5x^{2}-5x-17=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 5 mō a, -5 mō b, me -17 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\times 5\left(-17\right)}}{2\times 5}$$

Pūrua -5.

$$x=\frac{-\left(-5\right)±\sqrt{25-4\times 5\left(-17\right)}}{2\times 5}$$

Whakareatia -4 ki te 5.

$$x=\frac{-\left(-5\right)±\sqrt{25-20\left(-17\right)}}{2\times 5}$$

Whakareatia -20 ki te -17.

$$x=\frac{-\left(-5\right)±\sqrt{25+340}}{2\times 5}$$

Tāpiri 25 ki te 340.

$$x=\frac{-\left(-5\right)±\sqrt{365}}{2\times 5}$$

Ko te tauaro o -5 ko 5.

$$x=\frac{5±\sqrt{365}}{2\times 5}$$

Whakareatia 2 ki te 5.

$$x=\frac{5±\sqrt{365}}{10}$$

Nā, me whakaoti te whārite x=\frac{5±\sqrt{365}}{10} ina he tāpiri te ±. Tāpiri 5 ki te \sqrt{365}.

$$x=\frac{\sqrt{365}+5}{10}$$

Whakawehe 5+\sqrt{365} ki te 10.

$$x=\frac{\sqrt{365}}{10}+\frac{1}{2}$$

Nā, me whakaoti te whārite x=\frac{5±\sqrt{365}}{10} ina he tango te ±. Tango \sqrt{365} mai i 5.

$$x=\frac{5-\sqrt{365}}{10}$$

Whakawehe 5-\sqrt{365} ki te 10.

$$x=-\frac{\sqrt{365}}{10}+\frac{1}{2}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{365}}{10}+\frac{1}{2} x=-\frac{\sqrt{365}}{10}+\frac{1}{2}$$

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.

$$5x^{2}-5x-17=0$$

Me tāpiri 17 ki ngā taha e rua o te whārite.

$$5x^{2}-5x-17-\left(-17\right)=-\left(-17\right)$$

Mā te tango i te -17 i a ia ake anō ka toe ko te 0.

$$5x^{2}-5x=-\left(-17\right)$$

Tango -17 mai i 0.

$$5x^{2}-5x=17$$

Whakawehea ngā taha e rua ki te 5.

$$\frac{5x^{2}-5x}{5}=\frac{17}{5}$$

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

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

Whakawehe -5 ki te 5.

$$x^{2}-x=\frac{17}{5}$$

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.

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

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

$$x^{2}-x+\frac{1}{4}=\frac{17}{5}+\frac{1}{4}$$

Tāpiri \frac{17}{5} 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.

$$x^{2}-x+\frac{1}{4}=\frac{73}{20}$$

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}.

$$\left(x-\frac{1}{2}\right)^{2}=\frac{73}{20}$$

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

$$\sqrt{\left(x-\frac{1}{2}\right)^{2}}=\sqrt{\frac{73}{20}}$$

Whakarūnātia.

$$x-\frac{1}{2}=\frac{\sqrt{365}}{10} x-\frac{1}{2}=-\frac{\sqrt{365}}{10}$$

Me tāpiri \frac{1}{2} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{365}}{10}+\frac{1}{2} x=-\frac{\sqrt{365}}{10}+\frac{1}{2}$$