Whakaoti i te 13/4x^2-4x-5=0

$$\frac { 13 } { 4 } x ^ { 2 } - 4 x - 5 = 0$$

Whakaoti mō x

$$x=-\frac{10}{13}\approx -0.769230769$$

$$x=2$$

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.

$$\frac{13}{4}x^{2}-4x-5=0$$

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

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

Pūrua -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\times \frac{13}{4}\left(-5\right)}}{2\times \frac{13}{4}}$$

Whakareatia -4 ki te \frac{13}{4}.

$$x=\frac{-\left(-4\right)±\sqrt{16-13\left(-5\right)}}{2\times \frac{13}{4}}$$

Whakareatia -13 ki te -5.

$$x=\frac{-\left(-4\right)±\sqrt{16+65}}{2\times \frac{13}{4}}$$

Tāpiri 16 ki te 65.

$$x=\frac{-\left(-4\right)±\sqrt{81}}{2\times \frac{13}{4}}$$

Tuhia te pūtakerua o te 81.

$$x=\frac{-\left(-4\right)±9}{2\times \frac{13}{4}}$$

Ko te tauaro o -4 ko 4.

$$x=\frac{4±9}{2\times \frac{13}{4}}$$

Whakareatia 2 ki te \frac{13}{4}.

$$x=\frac{4±9}{\frac{13}{2}}$$

Nā, me whakaoti te whārite x=\frac{4±9}{\frac{13}{2}} ina he tāpiri te ±. Tāpiri 4 ki te 9.

$$x=\frac{13}{\frac{13}{2}}$$

Whakawehe 13 ki te \frac{13}{2} mā te whakarea 13 ki te tau huripoki o \frac{13}{2}.

$$x=2$$

Nā, me whakaoti te whārite x=\frac{4±9}{\frac{13}{2}} ina he tango te ±. Tango 9 mai i 4.

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

Whakawehe -5 ki te \frac{13}{2} mā te whakarea -5 ki te tau huripoki o \frac{13}{2}.

$$x=-\frac{10}{13}$$

Kua oti te whārite te whakatau.

$$x=2 x=-\frac{10}{13}$$

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.

$$\frac{13}{4}x^{2}-4x-5=0$$

Me tāpiri 5 ki ngā taha e rua o te whārite.

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

Mā te tango i te -5 i a ia ake anō ka toe ko te 0.

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

Tango -5 mai i 0.

$$\frac{13}{4}x^{2}-4x=5$$

Whakawehea ngā taha e rua o te whārite ki te \frac{13}{4}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$\frac{\frac{13}{4}x^{2}-4x}{\frac{13}{4}}=\frac{5}{\frac{13}{4}}$$

Mā te whakawehe ki te \frac{13}{4} ka wetekia te whakareanga ki te \frac{13}{4}.

$$x^{2}+\left(-\frac{4}{\frac{13}{4}}\right)x=\frac{5}{\frac{13}{4}}$$

Whakawehe -4 ki te \frac{13}{4} mā te whakarea -4 ki te tau huripoki o \frac{13}{4}.

$$x^{2}-\frac{16}{13}x=\frac{5}{\frac{13}{4}}$$

Whakawehe 5 ki te \frac{13}{4} mā te whakarea 5 ki te tau huripoki o \frac{13}{4}.

$$x^{2}-\frac{16}{13}x=\frac{20}{13}$$

Whakawehea te -\frac{16}{13}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{8}{13}. Nā, tāpiria te pūrua o te -\frac{8}{13} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{16}{13}x+\left(-\frac{8}{13}\right)^{2}=\frac{20}{13}+\left(-\frac{8}{13}\right)^{2}$$

Pūruatia -\frac{8}{13} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{16}{13}x+\frac{64}{169}=\frac{20}{13}+\frac{64}{169}$$

Tāpiri \frac{20}{13} ki te \frac{64}{169} 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.

$$x^{2}-\frac{16}{13}x+\frac{64}{169}=\frac{324}{169}$$

Tauwehea x^{2}-\frac{16}{13}x+\frac{64}{169}. 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}.

$$\left(x-\frac{8}{13}\right)^{2}=\frac{324}{169}$$

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

$$\sqrt{\left(x-\frac{8}{13}\right)^{2}}=\sqrt{\frac{324}{169}}$$

Whakarūnātia.

$$x-\frac{8}{13}=\frac{18}{13} x-\frac{8}{13}=-\frac{18}{13}$$

Me tāpiri \frac{8}{13} ki ngā taha e rua o te whārite.

$$x=2 x=-\frac{10}{13}$$