Whakaoti i te 3x^2+17x+10=0

$$3 { x }^{ 2 } +17x+10=0$$

Whakaoti mō x

$$x=-5$$

$$x=-\frac{2}{3}\approx -0.666666667$$

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 3x^{2}+ax+bx+10. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=17 ab=3\times 10=30$$

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 17.

$$a=2 b=15$$

Tuhia anō te 3x^{2}+17x+10 hei \left(3x^{2}+2x\right)+\left(15x+10\right).

$$\left(3x^{2}+2x\right)+\left(15x+10\right)$$

Tauwehea te x i te tuatahi me te 5 i te rōpū tuarua.

$$x\left(3x+2\right)+5\left(3x+2\right)$$

Whakatauwehea atu te kīanga pātahi 3x+2 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(3x+2\right)\left(x+5\right)$$

Hei kimi otinga whārite, me whakaoti te 3x+2=0 me te x+5=0.

$$x=-\frac{2}{3} x=-5$$

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.

$$3x^{2}+17x+10=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 3 mō a, 17 mō b, me 10 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-17±\sqrt{17^{2}-4\times 3\times 10}}{2\times 3}$$

Pūrua 17.

$$x=\frac{-17±\sqrt{289-4\times 3\times 10}}{2\times 3}$$

Whakareatia -4 ki te 3.

$$x=\frac{-17±\sqrt{289-12\times 10}}{2\times 3}$$

Whakareatia -12 ki te 10.

$$x=\frac{-17±\sqrt{289-120}}{2\times 3}$$

Tāpiri 289 ki te -120.

$$x=\frac{-17±\sqrt{169}}{2\times 3}$$

Tuhia te pūtakerua o te 169.

$$x=\frac{-17±13}{2\times 3}$$

Whakareatia 2 ki te 3.

$$x=\frac{-17±13}{6}$$

Nā, me whakaoti te whārite x=\frac{-17±13}{6} ina he tāpiri te ±. Tāpiri -17 ki te 13.

$$x=-\frac{4}{6}$$

Whakahekea te hautanga \frac{-4}{6} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x=-\frac{2}{3}$$

Nā, me whakaoti te whārite x=\frac{-17±13}{6} ina he tango te ±. Tango 13 mai i -17.

$$x=-\frac{30}{6}$$

Whakawehe -30 ki te 6.

$$x=-5$$

Kua oti te whārite te whakatau.

$$x=-\frac{2}{3} x=-5$$

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.

$$3x^{2}+17x+10=0$$

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

$$3x^{2}+17x+10-10=-10$$

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

$$3x^{2}+17x=-10$$

Whakawehea ngā taha e rua ki te 3.

$$\frac{3x^{2}+17x}{3}=-\frac{10}{3}$$

Mā te whakawehe ki te 3 ka wetekia te whakareanga ki te 3.

$$x^{2}+\frac{17}{3}x=-\frac{10}{3}$$

Whakawehea te \frac{17}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{17}{6}. Nā, tāpiria te pūrua o te \frac{17}{6} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{17}{3}x+\left(\frac{17}{6}\right)^{2}=-\frac{10}{3}+\left(\frac{17}{6}\right)^{2}$$

Pūruatia \frac{17}{6} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{17}{3}x+\frac{289}{36}=-\frac{10}{3}+\frac{289}{36}$$

Tāpiri -\frac{10}{3} ki te \frac{289}{36} 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.

$$x^{2}+\frac{17}{3}x+\frac{289}{36}=\frac{169}{36}$$

Tauwehea te x^{2}+\frac{17}{3}x+\frac{289}{36}. 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}.

$$\left(x+\frac{17}{6}\right)^{2}=\frac{169}{36}$$

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

$$\sqrt{\left(x+\frac{17}{6}\right)^{2}}=\sqrt{\frac{169}{36}}$$

Whakarūnātia.

$$x+\frac{17}{6}=\frac{13}{6} x+\frac{17}{6}=-\frac{13}{6}$$

Me tango \frac{17}{6} mai i ngā taha e rua o te whārite.

$$x=-\frac{2}{3} x=-5$$