Whakaoti i te 5x^2+21x=-10x-6

$$5 x ^ { 2 } + 21 x = - 10 x - 6$$

Whakaoti mō x

$$x=-6$$

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

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Me tāpiri te 10x ki ngā taha e rua.

$$5x^{2}+21x+10x=-6$$

Pahekotia te 21x me 10x, ka 31x.

$$5x^{2}+31x=-6$$

Me tāpiri te 6 ki ngā taha e rua.

$$5x^{2}+31x+6=0$$

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

$$a+b=31 ab=5\times 6=30$$

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 31.

$$a=1 b=30$$

Tuhia anō te 5x^{2}+31x+6 hei \left(5x^{2}+x\right)+\left(30x+6\right).

$$\left(5x^{2}+x\right)+\left(30x+6\right)$$

Tauwehea te x i te tuatahi me te 6 i te rōpū tuarua.

$$x\left(5x+1\right)+6\left(5x+1\right)$$

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

$$\left(5x+1\right)\left(x+6\right)$$

Hei kimi otinga whārite, me whakaoti te 5x+1=0 me te x+6=0.

$$x=-\frac{1}{5} x=-6$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Me tāpiri te 10x ki ngā taha e rua.

$$5x^{2}+21x+10x=-6$$

Pahekotia te 21x me 10x, ka 31x.

$$5x^{2}+31x=-6$$

Me tāpiri te 6 ki ngā taha e rua.

$$5x^{2}+31x+6=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 5 mō a, 31 mō b, me 6 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-31±\sqrt{31^{2}-4\times 5\times 6}}{2\times 5}$$

Pūrua 31.

$$x=\frac{-31±\sqrt{961-4\times 5\times 6}}{2\times 5}$$

Whakareatia -4 ki te 5.

$$x=\frac{-31±\sqrt{961-20\times 6}}{2\times 5}$$

Whakareatia -20 ki te 6.

$$x=\frac{-31±\sqrt{961-120}}{2\times 5}$$

Tāpiri 961 ki te -120.

$$x=\frac{-31±\sqrt{841}}{2\times 5}$$

Tuhia te pūtakerua o te 841.

$$x=\frac{-31±29}{2\times 5}$$

Whakareatia 2 ki te 5.

$$x=\frac{-31±29}{10}$$

Nā, me whakaoti te whārite x=\frac{-31±29}{10} ina he tāpiri te ±. Tāpiri -31 ki te 29.

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

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{-31±29}{10} ina he tango te ±. Tango 29 mai i -31.

$$x=-\frac{60}{10}$$

Whakawehe -60 ki te 10.

$$x=-6$$

Kua oti te whārite te whakatau.

$$x=-\frac{1}{5} x=-6$$

Ngā Upane mō te Whakaoti i te Pūrua

Me tāpiri te 10x ki ngā taha e rua.

$$5x^{2}+21x+10x=-6$$

Pahekotia te 21x me 10x, ka 31x.

$$5x^{2}+31x=-6$$

Whakawehea ngā taha e rua ki te 5.

$$\frac{5x^{2}+31x}{5}=-\frac{6}{5}$$

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

$$x^{2}+\frac{31}{5}x=-\frac{6}{5}$$

Whakawehea te \frac{31}{5}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{31}{10}. Nā, tāpiria te pūrua o te \frac{31}{10} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{31}{5}x+\left(\frac{31}{10}\right)^{2}=-\frac{6}{5}+\left(\frac{31}{10}\right)^{2}$$

Pūruatia \frac{31}{10} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{31}{5}x+\frac{961}{100}=-\frac{6}{5}+\frac{961}{100}$$

Tāpiri -\frac{6}{5} ki te \frac{961}{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.

$$x^{2}+\frac{31}{5}x+\frac{961}{100}=\frac{841}{100}$$

Tauwehea te x^{2}+\frac{31}{5}x+\frac{961}{100}. Ko te tikanga, ina ko x^{2}+bx+c he pūrua tika, ka taea te tauwehe i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{31}{10}\right)^{2}=\frac{841}{100}$$

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

$$\sqrt{\left(x+\frac{31}{10}\right)^{2}}=\sqrt{\frac{841}{100}}$$

Whakarūnātia.

$$x+\frac{31}{10}=\frac{29}{10} x+\frac{31}{10}=-\frac{29}{10}$$

Me tango \frac{31}{10} mai i ngā taha e rua o te whārite.

$$x=-\frac{1}{5} x=-6$$