Whakaoti i te 5x^2+21x-76=13

$$5 { x }^{ 2 } +21x-76=13$$

Whakaoti mō x

$$x = \frac{\sqrt{2221} - 21}{10} \approx 2.612748667$$

$$x=\frac{-\sqrt{2221}-21}{10}\approx -6.812748667$$

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.

$$5x^{2}+21x-76=13$$

Me tango 13 mai i ngā taha e rua o te whārite.

$$5x^{2}+21x-76-13=13-13$$

Mā te tango i te 13 i a ia ake anō ka toe ko te 0.

$$5x^{2}+21x-76-13=0$$

Tango 13 mai i -76.

$$5x^{2}+21x-89=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 5 mō a, 21 mō b, me -89 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-21±\sqrt{21^{2}-4\times 5\left(-89\right)}}{2\times 5}$$

Pūrua 21.

$$x=\frac{-21±\sqrt{441-4\times 5\left(-89\right)}}{2\times 5}$$

Whakareatia -4 ki te 5.

$$x=\frac{-21±\sqrt{441-20\left(-89\right)}}{2\times 5}$$

Whakareatia -20 ki te -89.

$$x=\frac{-21±\sqrt{441+1780}}{2\times 5}$$

Tāpiri 441 ki te 1780.

$$x=\frac{-21±\sqrt{2221}}{2\times 5}$$

Whakareatia 2 ki te 5.

$$x=\frac{-21±\sqrt{2221}}{10}$$

Nā, me whakaoti te whārite x=\frac{-21±\sqrt{2221}}{10} ina he tāpiri te ±. Tāpiri -21 ki te \sqrt{2221}.

$$x=\frac{\sqrt{2221}-21}{10}$$

Nā, me whakaoti te whārite x=\frac{-21±\sqrt{2221}}{10} ina he tango te ±. Tango \sqrt{2221} mai i -21.

$$x=\frac{-\sqrt{2221}-21}{10}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{2221}-21}{10} x=\frac{-\sqrt{2221}-21}{10}$$

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.

$$5x^{2}+21x-76=13$$

Me tāpiri 76 ki ngā taha e rua o te whārite.

$$5x^{2}+21x-76-\left(-76\right)=13-\left(-76\right)$$

Mā te tango i te -76 i a ia ake anō ka toe ko te 0.

$$5x^{2}+21x=13-\left(-76\right)$$

Tango -76 mai i 13.

$$5x^{2}+21x=89$$

Whakawehea ngā taha e rua ki te 5.

$$\frac{5x^{2}+21x}{5}=\frac{89}{5}$$

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

$$x^{2}+\frac{21}{5}x=\frac{89}{5}$$

Whakawehea te \frac{21}{5}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{21}{10}. Nā, tāpiria te pūrua o te \frac{21}{10} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{21}{5}x+\left(\frac{21}{10}\right)^{2}=\frac{89}{5}+\left(\frac{21}{10}\right)^{2}$$

Pūruatia \frac{21}{10} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{21}{5}x+\frac{441}{100}=\frac{89}{5}+\frac{441}{100}$$

Tāpiri \frac{89}{5} ki te \frac{441}{100} 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.

$$x^{2}+\frac{21}{5}x+\frac{441}{100}=\frac{2221}{100}$$

Tauwehea x^{2}+\frac{21}{5}x+\frac{441}{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}.

$$\left(x+\frac{21}{10}\right)^{2}=\frac{2221}{100}$$

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

$$\sqrt{\left(x+\frac{21}{10}\right)^{2}}=\sqrt{\frac{2221}{100}}$$

Whakarūnātia.

$$x+\frac{21}{10}=\frac{\sqrt{2221}}{10} x+\frac{21}{10}=-\frac{\sqrt{2221}}{10}$$

Me tango \frac{21}{10} mai i ngā taha e rua o te whārite.

$$x=\frac{\sqrt{2221}-21}{10} x=\frac{-\sqrt{2221}-21}{10}$$