Whakaoti i te 5w^2+13w=-6

$$5 w ^ { 2 } + 13 w = - 6$$

Whakaoti mō w

$$w=-\frac{3}{5}=-0.6$$

$$w=-2$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Me tāpiri te 6 ki ngā taha e rua.

$$5w^{2}+13w+6=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 5w^{2}+aw+bw+6. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=13 ab=5\times 6=30$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōrunga te a+b, he tōrunga hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 30.

$$1,30 2,15 3,10 5,6$$

Tātaihia te tapeke mō ia takirua.

$$1+30=31 2+15=17 3+10=13 5+6=11$$

Ko te otinga te takirua ka hoatu i te tapeke 13.

$$a=3 b=10$$

Tuhia anō te 5w^{2}+13w+6 hei \left(5w^{2}+3w\right)+\left(10w+6\right).

$$\left(5w^{2}+3w\right)+\left(10w+6\right)$$

Tauwehea te w i te tuatahi me te 2 i te rōpū tuarua.

$$w\left(5w+3\right)+2\left(5w+3\right)$$

Whakatauwehea atu te kīanga pātahi 5w+3 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(5w+3\right)\left(w+2\right)$$

Hei kimi otinga whārite, me whakaoti te 5w+3=0 me te w+2=0.

$$w=-\frac{3}{5} w=-2$$

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.

$$5w^{2}+13w=-6$$

Me tāpiri 6 ki ngā taha e rua o te whārite.

$$5w^{2}+13w-\left(-6\right)=-6-\left(-6\right)$$

Mā te tango i te -6 i a ia ake anō ka toe ko te 0.

$$5w^{2}+13w-\left(-6\right)=0$$

Tango -6 mai i 0.

$$5w^{2}+13w+6=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 5 mō a, 13 mō b, me 6 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$w=\frac{-13±\sqrt{13^{2}-4\times 5\times 6}}{2\times 5}$$

Pūrua 13.

$$w=\frac{-13±\sqrt{169-4\times 5\times 6}}{2\times 5}$$

Whakareatia -4 ki te 5.

$$w=\frac{-13±\sqrt{169-20\times 6}}{2\times 5}$$

Whakareatia -20 ki te 6.

$$w=\frac{-13±\sqrt{169-120}}{2\times 5}$$

Tāpiri 169 ki te -120.

$$w=\frac{-13±\sqrt{49}}{2\times 5}$$

Tuhia te pūtakerua o te 49.

$$w=\frac{-13±7}{2\times 5}$$

Whakareatia 2 ki te 5.

$$w=\frac{-13±7}{10}$$

Nā, me whakaoti te whārite w=\frac{-13±7}{10} ina he tāpiri te ±. Tāpiri -13 ki te 7.

$$w=-\frac{6}{10}$$

Whakahekea te hautanga \frac{-6}{10} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$w=-\frac{3}{5}$$

Nā, me whakaoti te whārite w=\frac{-13±7}{10} ina he tango te ±. Tango 7 mai i -13.

$$w=-\frac{20}{10}$$

Whakawehe -20 ki te 10.

$$w=-2$$

Kua oti te whārite te whakatau.

$$w=-\frac{3}{5} w=-2$$

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.

$$5w^{2}+13w=-6$$

Whakawehea ngā taha e rua ki te 5.

$$\frac{5w^{2}+13w}{5}=-\frac{6}{5}$$

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

$$w^{2}+\frac{13}{5}w=-\frac{6}{5}$$

Whakawehea te \frac{13}{5}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{13}{10}. Nā, tāpiria te pūrua o te \frac{13}{10} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$w^{2}+\frac{13}{5}w+\left(\frac{13}{10}\right)^{2}=-\frac{6}{5}+\left(\frac{13}{10}\right)^{2}$$

Pūruatia \frac{13}{10} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$w^{2}+\frac{13}{5}w+\frac{169}{100}=-\frac{6}{5}+\frac{169}{100}$$

Tāpiri -\frac{6}{5} ki te \frac{169}{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.

$$w^{2}+\frac{13}{5}w+\frac{169}{100}=\frac{49}{100}$$

Tauwehea w^{2}+\frac{13}{5}w+\frac{169}{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}.

$$\left(w+\frac{13}{10}\right)^{2}=\frac{49}{100}$$

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

$$\sqrt{\left(w+\frac{13}{10}\right)^{2}}=\sqrt{\frac{49}{100}}$$

Whakarūnātia.

$$w+\frac{13}{10}=\frac{7}{10} w+\frac{13}{10}=-\frac{7}{10}$$

Me tango \frac{13}{10} mai i ngā taha e rua o te whārite.

$$w=-\frac{3}{5} w=-2$$