Whakaoti i te 5x^2+21x+4=0

$$5 x ^ { 2 } + 21 x + 4 = 0$$

Whakaoti mō x

$$x=-4$$

$$x=-\frac{1}{5}=-0.2$$

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 5x^{2}+ax+bx+4. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=21 ab=5\times 4=20$$

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 20.

$$1,20 2,10 4,5$$

Tātaihia te tapeke mō ia takirua.

$$1+20=21 2+10=12 4+5=9$$

Ko te otinga te takirua ka hoatu i te tapeke 21.

$$a=1 b=20$$

Tuhia anō te 5x^{2}+21x+4 hei \left(5x^{2}+x\right)+\left(20x+4\right).

$$\left(5x^{2}+x\right)+\left(20x+4\right)$$

Tauwehea te x i te tuatahi me te 4 i te rōpū tuarua.

$$x\left(5x+1\right)+4\left(5x+1\right)$$

Whakatauwehea atu te kīanga pātahi 5x+1 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(5x+1\right)\left(x+4\right)$$

Hei kimi otinga whārite, me whakaoti te 5x+1=0 me te x+4=0.

$$x=-\frac{1}{5} x=-4$$

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.

$$5x^{2}+21x+4=0$$

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

$$x=\frac{-21±\sqrt{21^{2}-4\times 5\times 4}}{2\times 5}$$

Pūrua 21.

$$x=\frac{-21±\sqrt{441-4\times 5\times 4}}{2\times 5}$$

Whakareatia -4 ki te 5.

$$x=\frac{-21±\sqrt{441-20\times 4}}{2\times 5}$$

Whakareatia -20 ki te 4.

$$x=\frac{-21±\sqrt{441-80}}{2\times 5}$$

Tāpiri 441 ki te -80.

$$x=\frac{-21±\sqrt{361}}{2\times 5}$$

Tuhia te pūtakerua o te 361.

$$x=\frac{-21±19}{2\times 5}$$

Whakareatia 2 ki te 5.

$$x=\frac{-21±19}{10}$$

Nā, me whakaoti te whārite x=\frac{-21±19}{10} ina he tāpiri te ±. Tāpiri -21 ki te 19.

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

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

$$x=-\frac{1}{5}$$

Nā, me whakaoti te whārite x=\frac{-21±19}{10} ina he tango te ±. Tango 19 mai i -21.

$$x=-\frac{40}{10}$$

Whakawehe -40 ki te 10.

$$x=-4$$

Kua oti te whārite te whakatau.

$$x=-\frac{1}{5} x=-4$$

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.

$$5x^{2}+21x+4=0$$

Me tango 4 mai i ngā taha e rua o te whārite.

$$5x^{2}+21x+4-4=-4$$

Mā te tango i te 4 i a ia ake anō ka toe ko te 0.

$$5x^{2}+21x=-4$$

Whakawehea ngā taha e rua ki te 5.

$$\frac{5x^{2}+21x}{5}=-\frac{4}{5}$$

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

$$x^{2}+\frac{21}{5}x=-\frac{4}{5}$$

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.

$$x^{2}+\frac{21}{5}x+\left(\frac{21}{10}\right)^{2}=-\frac{4}{5}+\left(\frac{21}{10}\right)^{2}$$

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

$$x^{2}+\frac{21}{5}x+\frac{441}{100}=-\frac{4}{5}+\frac{441}{100}$$

Tāpiri -\frac{4}{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.

$$x^{2}+\frac{21}{5}x+\frac{441}{100}=\frac{361}{100}$$

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}.

$$\left(x+\frac{21}{10}\right)^{2}=\frac{361}{100}$$

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

$$\sqrt{\left(x+\frac{21}{10}\right)^{2}}=\sqrt{\frac{361}{100}}$$

Whakarūnātia.

$$x+\frac{21}{10}=\frac{19}{10} x+\frac{21}{10}=-\frac{19}{10}$$

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

$$x=-\frac{1}{5} x=-4$$