Whakaoti i te 10k^2+9k-1=0

$$10 k ^ { 2 } + 9 k - 1 = 0$$

Whakaoti mō k

$$k=-1$$

$$k=\frac{1}{10}=0.1$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 10k^{2}+ak+bk-1. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=9 ab=10\left(-1\right)=-10$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōrunga te a+b, he nui ake te uara pū o te tau tōrunga i tō te tōraro. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -10.

$$-1,10 -2,5$$

Tātaihia te tapeke mō ia takirua.

$$-1+10=9 -2+5=3$$

Ko te otinga te takirua ka hoatu i te tapeke 9.

$$a=-1 b=10$$

Tuhia anō te 10k^{2}+9k-1 hei \left(10k^{2}-k\right)+\left(10k-1\right).

$$\left(10k^{2}-k\right)+\left(10k-1\right)$$

Whakatauwehea atu k i te 10k^{2}-k.

$$k\left(10k-1\right)+10k-1$$

Whakatauwehea atu te kīanga pātahi 10k-1 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(10k-1\right)\left(k+1\right)$$

Hei kimi otinga whārite, me whakaoti te 10k-1=0 me te k+1=0.

$$k=\frac{1}{10} k=-1$$

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.

$$10k^{2}+9k-1=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 10 mō a, 9 mō b, me -1 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$k=\frac{-9±\sqrt{9^{2}-4\times 10\left(-1\right)}}{2\times 10}$$

Pūrua 9.

$$k=\frac{-9±\sqrt{81-4\times 10\left(-1\right)}}{2\times 10}$$

Whakareatia -4 ki te 10.

$$k=\frac{-9±\sqrt{81-40\left(-1\right)}}{2\times 10}$$

Whakareatia -40 ki te -1.

$$k=\frac{-9±\sqrt{81+40}}{2\times 10}$$

Tāpiri 81 ki te 40.

$$k=\frac{-9±\sqrt{121}}{2\times 10}$$

Tuhia te pūtakerua o te 121.

$$k=\frac{-9±11}{2\times 10}$$

Whakareatia 2 ki te 10.

$$k=\frac{-9±11}{20}$$

Nā, me whakaoti te whārite k=\frac{-9±11}{20} ina he tāpiri te ±. Tāpiri -9 ki te 11.

$$k=\frac{2}{20}$$

Whakahekea te hautanga \frac{2}{20} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$k=\frac{1}{10}$$

Nā, me whakaoti te whārite k=\frac{-9±11}{20} ina he tango te ±. Tango 11 mai i -9.

$$k=-\frac{20}{20}$$

Whakawehe -20 ki te 20.

$$k=-1$$

Kua oti te whārite te whakatau.

$$k=\frac{1}{10} k=-1$$

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.

$$10k^{2}+9k-1=0$$

Me tāpiri 1 ki ngā taha e rua o te whārite.

$$10k^{2}+9k-1-\left(-1\right)=-\left(-1\right)$$

Mā te tango i te -1 i a ia ake anō ka toe ko te 0.

$$10k^{2}+9k=-\left(-1\right)$$

Tango -1 mai i 0.

$$10k^{2}+9k=1$$

Whakawehea ngā taha e rua ki te 10.

$$\frac{10k^{2}+9k}{10}=\frac{1}{10}$$

Mā te whakawehe ki te 10 ka wetekia te whakareanga ki te 10.

$$k^{2}+\frac{9}{10}k=\frac{1}{10}$$

Whakawehea te \frac{9}{10}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{9}{20}. Nā, tāpiria te pūrua o te \frac{9}{20} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$k^{2}+\frac{9}{10}k+\left(\frac{9}{20}\right)^{2}=\frac{1}{10}+\left(\frac{9}{20}\right)^{2}$$

Pūruatia \frac{9}{20} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$k^{2}+\frac{9}{10}k+\frac{81}{400}=\frac{1}{10}+\frac{81}{400}$$

Tāpiri \frac{1}{10} ki te \frac{81}{400} 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.

$$k^{2}+\frac{9}{10}k+\frac{81}{400}=\frac{121}{400}$$

Tauwehea k^{2}+\frac{9}{10}k+\frac{81}{400}. 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}.

$$\left(k+\frac{9}{20}\right)^{2}=\frac{121}{400}$$

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

$$\sqrt{\left(k+\frac{9}{20}\right)^{2}}=\sqrt{\frac{121}{400}}$$

Whakarūnātia.

$$k+\frac{9}{20}=\frac{11}{20} k+\frac{9}{20}=-\frac{11}{20}$$

Me tango \frac{9}{20} mai i ngā taha e rua o te whārite.

$$k=\frac{1}{10} k=-1$$