Whakaoti i te 6x^2+17x+10=0

$$6 x ^ { 2 } + 17 x + 10 = 0$$

Whakaoti mō x

$$x=-\frac{5}{6}\approx -0.833333333$$

$$x=-2$$

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 6x^{2}+ax+bx+10. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=17 ab=6\times 10=60$$

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 60.

$$1,60 2,30 3,20 4,15 5,12 6,10$$

Tātaihia te tapeke mō ia takirua.

$$1+60=61 2+30=32 3+20=23 4+15=19 5+12=17 6+10=16$$

Ko te otinga te takirua ka hoatu i te tapeke 17.

$$a=5 b=12$$

Tuhia anō te 6x^{2}+17x+10 hei \left(6x^{2}+5x\right)+\left(12x+10\right).

$$\left(6x^{2}+5x\right)+\left(12x+10\right)$$

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

$$x\left(6x+5\right)+2\left(6x+5\right)$$

Whakatauwehea atu te kīanga pātahi 6x+5 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(6x+5\right)\left(x+2\right)$$

Hei kimi otinga whārite, me whakaoti te 6x+5=0 me te x+2=0.

$$x=-\frac{5}{6} x=-2$$

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.

$$6x^{2}+17x+10=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 6 mō a, 17 mō b, me 10 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-17±\sqrt{17^{2}-4\times 6\times 10}}{2\times 6}$$

Pūrua 17.

$$x=\frac{-17±\sqrt{289-4\times 6\times 10}}{2\times 6}$$

Whakareatia -4 ki te 6.

$$x=\frac{-17±\sqrt{289-24\times 10}}{2\times 6}$$

Whakareatia -24 ki te 10.

$$x=\frac{-17±\sqrt{289-240}}{2\times 6}$$

Tāpiri 289 ki te -240.

$$x=\frac{-17±\sqrt{49}}{2\times 6}$$

Tuhia te pūtakerua o te 49.

$$x=\frac{-17±7}{2\times 6}$$

Whakareatia 2 ki te 6.

$$x=\frac{-17±7}{12}$$

Nā, me whakaoti te whārite x=\frac{-17±7}{12} ina he tāpiri te ±. Tāpiri -17 ki te 7.

$$x=-\frac{10}{12}$$

Whakahekea te hautanga \frac{-10}{12} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

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

Nā, me whakaoti te whārite x=\frac{-17±7}{12} ina he tango te ±. Tango 7 mai i -17.

$$x=-\frac{24}{12}$$

Whakawehe -24 ki te 12.

$$x=-2$$

Kua oti te whārite te whakatau.

$$x=-\frac{5}{6} x=-2$$

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.

$$6x^{2}+17x+10=0$$

Me tango 10 mai i ngā taha e rua o te whārite.

$$6x^{2}+17x+10-10=-10$$

Mā te tango i te 10 i a ia ake anō ka toe ko te 0.

$$6x^{2}+17x=-10$$

Whakawehea ngā taha e rua ki te 6.

$$\frac{6x^{2}+17x}{6}=-\frac{10}{6}$$

Mā te whakawehe ki te 6 ka wetekia te whakareanga ki te 6.

$$x^{2}+\frac{17}{6}x=-\frac{10}{6}$$

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

$$x^{2}+\frac{17}{6}x=-\frac{5}{3}$$

Whakawehea te \frac{17}{6}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{17}{12}. Nā, tāpiria te pūrua o te \frac{17}{12} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{17}{6}x+\left(\frac{17}{12}\right)^{2}=-\frac{5}{3}+\left(\frac{17}{12}\right)^{2}$$

Pūruatia \frac{17}{12} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{17}{6}x+\frac{289}{144}=-\frac{5}{3}+\frac{289}{144}$$

Tāpiri -\frac{5}{3} ki te \frac{289}{144} 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.

$$x^{2}+\frac{17}{6}x+\frac{289}{144}=\frac{49}{144}$$

Tauwehea x^{2}+\frac{17}{6}x+\frac{289}{144}. 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}.

$$\left(x+\frac{17}{12}\right)^{2}=\frac{49}{144}$$

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

$$\sqrt{\left(x+\frac{17}{12}\right)^{2}}=\sqrt{\frac{49}{144}}$$

Whakarūnātia.

$$x+\frac{17}{12}=\frac{7}{12} x+\frac{17}{12}=-\frac{7}{12}$$

Me tango \frac{17}{12} mai i ngā taha e rua o te whārite.

$$x=-\frac{5}{6} x=-2$$