Whakaoti i te -110=37587x+-491x^2

$$-110 = 37587x+-491 { x }^{ 2 }$$

Whakaoti mō x

$$x = \frac{\sqrt{1412998609} + 37587}{982} \approx 76.554861259$$

$$x=\frac{37587-\sqrt{1412998609}}{982}\approx -0.002926432$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$37587x-491x^{2}=-110$$

Me tāpiri te 110 ki ngā taha e rua.

$$37587x-491x^{2}+110=0$$

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.

$$-491x^{2}+37587x+110=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -491 mō a, 37587 mō b, me 110 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-37587±\sqrt{37587^{2}-4\left(-491\right)\times 110}}{2\left(-491\right)}$$

Pūrua 37587.

$$x=\frac{-37587±\sqrt{1412782569-4\left(-491\right)\times 110}}{2\left(-491\right)}$$

Whakareatia -4 ki te -491.

$$x=\frac{-37587±\sqrt{1412782569+1964\times 110}}{2\left(-491\right)}$$

Whakareatia 1964 ki te 110.

$$x=\frac{-37587±\sqrt{1412782569+216040}}{2\left(-491\right)}$$

Tāpiri 1412782569 ki te 216040.

$$x=\frac{-37587±\sqrt{1412998609}}{2\left(-491\right)}$$

Whakareatia 2 ki te -491.

$$x=\frac{-37587±\sqrt{1412998609}}{-982}$$

Nā, me whakaoti te whārite x=\frac{-37587±\sqrt{1412998609}}{-982} ina he tāpiri te ±. Tāpiri -37587 ki te \sqrt{1412998609}.

$$x=\frac{\sqrt{1412998609}-37587}{-982}$$

Whakawehe -37587+\sqrt{1412998609} ki te -982.

$$x=\frac{37587-\sqrt{1412998609}}{982}$$

Nā, me whakaoti te whārite x=\frac{-37587±\sqrt{1412998609}}{-982} ina he tango te ±. Tango \sqrt{1412998609} mai i -37587.

$$x=\frac{-\sqrt{1412998609}-37587}{-982}$$

Whakawehe -37587-\sqrt{1412998609} ki te -982.

$$x=\frac{\sqrt{1412998609}+37587}{982}$$

Kua oti te whārite te whakatau.

$$x=\frac{37587-\sqrt{1412998609}}{982} x=\frac{\sqrt{1412998609}+37587}{982}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$37587x-491x^{2}=-110$$

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.

$$-491x^{2}+37587x=-110$$

Whakawehea ngā taha e rua ki te -491.

$$\frac{-491x^{2}+37587x}{-491}=-\frac{110}{-491}$$

Mā te whakawehe ki te -491 ka wetekia te whakareanga ki te -491.

$$x^{2}+\frac{37587}{-491}x=-\frac{110}{-491}$$

Whakawehe 37587 ki te -491.

$$x^{2}-\frac{37587}{491}x=-\frac{110}{-491}$$

Whakawehe -110 ki te -491.

$$x^{2}-\frac{37587}{491}x=\frac{110}{491}$$

Whakawehea te -\frac{37587}{491}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{37587}{982}. Nā, tāpiria te pūrua o te -\frac{37587}{982} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{37587}{491}x+\left(-\frac{37587}{982}\right)^{2}=\frac{110}{491}+\left(-\frac{37587}{982}\right)^{2}$$

Pūruatia -\frac{37587}{982} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{37587}{491}x+\frac{1412782569}{964324}=\frac{110}{491}+\frac{1412782569}{964324}$$

Tāpiri \frac{110}{491} ki te \frac{1412782569}{964324} 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.

$$x^{2}-\frac{37587}{491}x+\frac{1412782569}{964324}=\frac{1412998609}{964324}$$

Tauwehea x^{2}-\frac{37587}{491}x+\frac{1412782569}{964324}. 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}.

$$\left(x-\frac{37587}{982}\right)^{2}=\frac{1412998609}{964324}$$

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

$$\sqrt{\left(x-\frac{37587}{982}\right)^{2}}=\sqrt{\frac{1412998609}{964324}}$$

Whakarūnātia.

$$x-\frac{37587}{982}=\frac{\sqrt{1412998609}}{982} x-\frac{37587}{982}=-\frac{\sqrt{1412998609}}{982}$$

Me tāpiri \frac{37587}{982} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{1412998609}+37587}{982} x=\frac{37587-\sqrt{1412998609}}{982}$$