Whakaoti i te 24k^2+50k+24=0

$$24 { k }^{ 2 } +50k+24 = 0$$

Whakaoti mō k

$$k = -\frac{4}{3} = -1\frac{1}{3} \approx -1.333333333$$

$$k=-\frac{3}{4}=-0.75$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakawehea ngā taha e rua ki te 2.

$$12k^{2}+25k+12=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 12k^{2}+ak+bk+12. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=25 ab=12\times 12=144$$

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 144.

$$1,144 2,72 3,48 4,36 6,24 8,18 9,16 12,12$$

Tātaihia te tapeke mō ia takirua.

$$1+144=145 2+72=74 3+48=51 4+36=40 6+24=30 8+18=26 9+16=25 12+12=24$$

Ko te otinga te takirua ka hoatu i te tapeke 25.

$$a=9 b=16$$

Tuhia anō te 12k^{2}+25k+12 hei \left(12k^{2}+9k\right)+\left(16k+12\right).

$$\left(12k^{2}+9k\right)+\left(16k+12\right)$$

Tauwehea te 3k i te tuatahi me te 4 i te rōpū tuarua.

$$3k\left(4k+3\right)+4\left(4k+3\right)$$

Whakatauwehea atu te kīanga pātahi 4k+3 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(4k+3\right)\left(3k+4\right)$$

Hei kimi otinga whārite, me whakaoti te 4k+3=0 me te 3k+4=0.

$$k=-\frac{3}{4} k=-\frac{4}{3}$$

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.

$$24k^{2}+50k+24=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 24 mō a, 50 mō b, me 24 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$k=\frac{-50±\sqrt{50^{2}-4\times 24\times 24}}{2\times 24}$$

Pūrua 50.

$$k=\frac{-50±\sqrt{2500-4\times 24\times 24}}{2\times 24}$$

Whakareatia -4 ki te 24.

$$k=\frac{-50±\sqrt{2500-96\times 24}}{2\times 24}$$

Whakareatia -96 ki te 24.

$$k=\frac{-50±\sqrt{2500-2304}}{2\times 24}$$

Tāpiri 2500 ki te -2304.

$$k=\frac{-50±\sqrt{196}}{2\times 24}$$

Tuhia te pūtakerua o te 196.

$$k=\frac{-50±14}{2\times 24}$$

Whakareatia 2 ki te 24.

$$k=\frac{-50±14}{48}$$

Nā, me whakaoti te whārite k=\frac{-50±14}{48} ina he tāpiri te ±. Tāpiri -50 ki te 14.

$$k=-\frac{36}{48}$$

Whakahekea te hautanga \frac{-36}{48} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 12.

$$k=-\frac{3}{4}$$

Nā, me whakaoti te whārite k=\frac{-50±14}{48} ina he tango te ±. Tango 14 mai i -50.

$$k=-\frac{64}{48}$$

Whakahekea te hautanga \frac{-64}{48} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 16.

$$k=-\frac{4}{3}$$

Kua oti te whārite te whakatau.

$$k=-\frac{3}{4} k=-\frac{4}{3}$$

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.

$$24k^{2}+50k+24=0$$

Me tango 24 mai i ngā taha e rua o te whārite.

$$24k^{2}+50k+24-24=-24$$

Mā te tango i te 24 i a ia ake anō ka toe ko te 0.

$$24k^{2}+50k=-24$$

Whakawehea ngā taha e rua ki te 24.

$$\frac{24k^{2}+50k}{24}=-\frac{24}{24}$$

Mā te whakawehe ki te 24 ka wetekia te whakareanga ki te 24.

$$k^{2}+\frac{50}{24}k=-\frac{24}{24}$$

Whakahekea te hautanga \frac{50}{24} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$k^{2}+\frac{25}{12}k=-\frac{24}{24}$$

Whakawehe -24 ki te 24.

$$k^{2}+\frac{25}{12}k=-1$$

Whakawehea te \frac{25}{12}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{25}{24}. Nā, tāpiria te pūrua o te \frac{25}{24} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$k^{2}+\frac{25}{12}k+\left(\frac{25}{24}\right)^{2}=-1+\left(\frac{25}{24}\right)^{2}$$

Pūruatia \frac{25}{24} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$k^{2}+\frac{25}{12}k+\frac{625}{576}=-1+\frac{625}{576}$$

Tāpiri -1 ki te \frac{625}{576}.

$$k^{2}+\frac{25}{12}k+\frac{625}{576}=\frac{49}{576}$$

Tauwehea k^{2}+\frac{25}{12}k+\frac{625}{576}. 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}.

$$\left(k+\frac{25}{24}\right)^{2}=\frac{49}{576}$$

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

$$\sqrt{\left(k+\frac{25}{24}\right)^{2}}=\sqrt{\frac{49}{576}}$$

Whakarūnātia.

$$k+\frac{25}{24}=\frac{7}{24} k+\frac{25}{24}=-\frac{7}{24}$$

Me tango \frac{25}{24} mai i ngā taha e rua o te whārite.

$$k=-\frac{3}{4} k=-\frac{4}{3}$$