Whakaoti i te 3x^2-50x-26=0

$$3 { x }^{ 2 } -50x-26=0$$

Whakaoti mō x

$$x = \frac{\sqrt{703} + 25}{3} \approx 17.171382389$$

$$x=\frac{25-\sqrt{703}}{3}\approx -0.504715722$$

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.

$$3x^{2}-50x-26=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 3 mō a, -50 mō b, me -26 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-50\right)±\sqrt{\left(-50\right)^{2}-4\times 3\left(-26\right)}}{2\times 3}$$

Pūrua -50.

$$x=\frac{-\left(-50\right)±\sqrt{2500-4\times 3\left(-26\right)}}{2\times 3}$$

Whakareatia -4 ki te 3.

$$x=\frac{-\left(-50\right)±\sqrt{2500-12\left(-26\right)}}{2\times 3}$$

Whakareatia -12 ki te -26.

$$x=\frac{-\left(-50\right)±\sqrt{2500+312}}{2\times 3}$$

Tāpiri 2500 ki te 312.

$$x=\frac{-\left(-50\right)±\sqrt{2812}}{2\times 3}$$

Tuhia te pūtakerua o te 2812.

$$x=\frac{-\left(-50\right)±2\sqrt{703}}{2\times 3}$$

Ko te tauaro o -50 ko 50.

$$x=\frac{50±2\sqrt{703}}{2\times 3}$$

Whakareatia 2 ki te 3.

$$x=\frac{50±2\sqrt{703}}{6}$$

Nā, me whakaoti te whārite x=\frac{50±2\sqrt{703}}{6} ina he tāpiri te ±. Tāpiri 50 ki te 2\sqrt{703}.

$$x=\frac{2\sqrt{703}+50}{6}$$

Whakawehe 50+2\sqrt{703} ki te 6.

$$x=\frac{\sqrt{703}+25}{3}$$

Nā, me whakaoti te whārite x=\frac{50±2\sqrt{703}}{6} ina he tango te ±. Tango 2\sqrt{703} mai i 50.

$$x=\frac{50-2\sqrt{703}}{6}$$

Whakawehe 50-2\sqrt{703} ki te 6.

$$x=\frac{25-\sqrt{703}}{3}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{703}+25}{3} x=\frac{25-\sqrt{703}}{3}$$

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.

$$3x^{2}-50x-26=0$$

Me tāpiri 26 ki ngā taha e rua o te whārite.

$$3x^{2}-50x-26-\left(-26\right)=-\left(-26\right)$$

Mā te tango i te -26 i a ia ake anō ka toe ko te 0.

$$3x^{2}-50x=-\left(-26\right)$$

Tango -26 mai i 0.

$$3x^{2}-50x=26$$

Whakawehea ngā taha e rua ki te 3.

$$\frac{3x^{2}-50x}{3}=\frac{26}{3}$$

Mā te whakawehe ki te 3 ka wetekia te whakareanga ki te 3.

$$x^{2}-\frac{50}{3}x=\frac{26}{3}$$

Whakawehea te -\frac{50}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{25}{3}. Nā, tāpiria te pūrua o te -\frac{25}{3} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{50}{3}x+\left(-\frac{25}{3}\right)^{2}=\frac{26}{3}+\left(-\frac{25}{3}\right)^{2}$$

Pūruatia -\frac{25}{3} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{50}{3}x+\frac{625}{9}=\frac{26}{3}+\frac{625}{9}$$

Tāpiri \frac{26}{3} ki te \frac{625}{9} 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.

$$x^{2}-\frac{50}{3}x+\frac{625}{9}=\frac{703}{9}$$

Tauwehea te x^{2}-\frac{50}{3}x+\frac{625}{9}. Ko te tikanga, ina ko x^{2}+bx+c he pūrua tika, ka taea te tauwehe i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{25}{3}\right)^{2}=\frac{703}{9}$$

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

$$\sqrt{\left(x-\frac{25}{3}\right)^{2}}=\sqrt{\frac{703}{9}}$$

Whakarūnātia.

$$x-\frac{25}{3}=\frac{\sqrt{703}}{3} x-\frac{25}{3}=-\frac{\sqrt{703}}{3}$$

Me tāpiri \frac{25}{3} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{703}+25}{3} x=\frac{25-\sqrt{703}}{3}$$