Whakaoti i te 13x^2-5x-20=0

$$13 x ^ { 2 } - 5 x - 20 = 0$$

Whakaoti mō x

$$x = \frac{\sqrt{1065} + 5}{26} \approx 1.447474529$$

$$x=\frac{5-\sqrt{1065}}{26}\approx -1.062859144$$

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.

$$13x^{2}-5x-20=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 13 mō a, -5 mō b, me -20 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\times 13\left(-20\right)}}{2\times 13}$$

Pūrua -5.

$$x=\frac{-\left(-5\right)±\sqrt{25-4\times 13\left(-20\right)}}{2\times 13}$$

Whakareatia -4 ki te 13.

$$x=\frac{-\left(-5\right)±\sqrt{25-52\left(-20\right)}}{2\times 13}$$

Whakareatia -52 ki te -20.

$$x=\frac{-\left(-5\right)±\sqrt{25+1040}}{2\times 13}$$

Tāpiri 25 ki te 1040.

$$x=\frac{-\left(-5\right)±\sqrt{1065}}{2\times 13}$$

Ko te tauaro o -5 ko 5.

$$x=\frac{5±\sqrt{1065}}{2\times 13}$$

Whakareatia 2 ki te 13.

$$x=\frac{5±\sqrt{1065}}{26}$$

Nā, me whakaoti te whārite x=\frac{5±\sqrt{1065}}{26} ina he tāpiri te ±. Tāpiri 5 ki te \sqrt{1065}.

$$x=\frac{\sqrt{1065}+5}{26}$$

Nā, me whakaoti te whārite x=\frac{5±\sqrt{1065}}{26} ina he tango te ±. Tango \sqrt{1065} mai i 5.

$$x=\frac{5-\sqrt{1065}}{26}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{1065}+5}{26} x=\frac{5-\sqrt{1065}}{26}$$

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.

$$13x^{2}-5x-20=0$$

Me tāpiri 20 ki ngā taha e rua o te whārite.

$$13x^{2}-5x-20-\left(-20\right)=-\left(-20\right)$$

Mā te tango i te -20 i a ia ake anō ka toe ko te 0.

$$13x^{2}-5x=-\left(-20\right)$$

Tango -20 mai i 0.

$$13x^{2}-5x=20$$

Whakawehea ngā taha e rua ki te 13.

$$\frac{13x^{2}-5x}{13}=\frac{20}{13}$$

Mā te whakawehe ki te 13 ka wetekia te whakareanga ki te 13.

$$x^{2}-\frac{5}{13}x=\frac{20}{13}$$

Whakawehea te -\frac{5}{13}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{5}{26}. Nā, tāpiria te pūrua o te -\frac{5}{26} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{5}{13}x+\left(-\frac{5}{26}\right)^{2}=\frac{20}{13}+\left(-\frac{5}{26}\right)^{2}$$

Pūruatia -\frac{5}{26} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{5}{13}x+\frac{25}{676}=\frac{20}{13}+\frac{25}{676}$$

Tāpiri \frac{20}{13} ki te \frac{25}{676} 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.

$$x^{2}-\frac{5}{13}x+\frac{25}{676}=\frac{1065}{676}$$

Tauwehea x^{2}-\frac{5}{13}x+\frac{25}{676}. 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}.

$$\left(x-\frac{5}{26}\right)^{2}=\frac{1065}{676}$$

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

$$\sqrt{\left(x-\frac{5}{26}\right)^{2}}=\sqrt{\frac{1065}{676}}$$

Whakarūnātia.

$$x-\frac{5}{26}=\frac{\sqrt{1065}}{26} x-\frac{5}{26}=-\frac{\sqrt{1065}}{26}$$

Me tāpiri \frac{5}{26} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{1065}+5}{26} x=\frac{5-\sqrt{1065}}{26}$$