Whakaoti i te 39x^2+14x-9=0

$$39 { x }^{ 2 } +14x-9=0$$

Whakaoti mō x

$$x=-\frac{9}{13}\approx -0.692307692$$

$$x=\frac{1}{3}\approx 0.333333333$$

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 39x^{2}+ax+bx-9. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=14 ab=39\left(-9\right)=-351$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōrunga te a+b, he nui ake te uara pū o te tau tōrunga i tō te tōraro. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -351.

$$-1,351 -3,117 -9,39 -13,27$$

Tātaihia te tapeke mō ia takirua.

$$-1+351=350 -3+117=114 -9+39=30 -13+27=14$$

Ko te otinga te takirua ka hoatu i te tapeke 14.

$$a=-13 b=27$$

Tuhia anō te 39x^{2}+14x-9 hei \left(39x^{2}-13x\right)+\left(27x-9\right).

$$\left(39x^{2}-13x\right)+\left(27x-9\right)$$

Tauwehea te 13x i te tuatahi me te 9 i te rōpū tuarua.

$$13x\left(3x-1\right)+9\left(3x-1\right)$$

Whakatauwehea atu te kīanga pātahi 3x-1 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(3x-1\right)\left(13x+9\right)$$

Hei kimi otinga whārite, me whakaoti te 3x-1=0 me te 13x+9=0.

$$x=\frac{1}{3} x=-\frac{9}{13}$$

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.

$$39x^{2}+14x-9=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 39 mō a, 14 mō b, me -9 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-14±\sqrt{14^{2}-4\times 39\left(-9\right)}}{2\times 39}$$

Pūrua 14.

$$x=\frac{-14±\sqrt{196-4\times 39\left(-9\right)}}{2\times 39}$$

Whakareatia -4 ki te 39.

$$x=\frac{-14±\sqrt{196-156\left(-9\right)}}{2\times 39}$$

Whakareatia -156 ki te -9.

$$x=\frac{-14±\sqrt{196+1404}}{2\times 39}$$

Tāpiri 196 ki te 1404.

$$x=\frac{-14±\sqrt{1600}}{2\times 39}$$

Tuhia te pūtakerua o te 1600.

$$x=\frac{-14±40}{2\times 39}$$

Whakareatia 2 ki te 39.

$$x=\frac{-14±40}{78}$$

Nā, me whakaoti te whārite x=\frac{-14±40}{78} ina he tāpiri te ±. Tāpiri -14 ki te 40.

$$x=\frac{26}{78}$$

Whakahekea te hautanga \frac{26}{78} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 26.

$$x=\frac{1}{3}$$

Nā, me whakaoti te whārite x=\frac{-14±40}{78} ina he tango te ±. Tango 40 mai i -14.

$$x=-\frac{54}{78}$$

Whakahekea te hautanga \frac{-54}{78} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 6.

$$x=-\frac{9}{13}$$

Kua oti te whārite te whakatau.

$$x=\frac{1}{3} x=-\frac{9}{13}$$

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.

$$39x^{2}+14x-9=0$$

Me tāpiri 9 ki ngā taha e rua o te whārite.

$$39x^{2}+14x-9-\left(-9\right)=-\left(-9\right)$$

Mā te tango i te -9 i a ia ake anō ka toe ko te 0.

$$39x^{2}+14x=-\left(-9\right)$$

Tango -9 mai i 0.

$$39x^{2}+14x=9$$

Whakawehea ngā taha e rua ki te 39.

$$\frac{39x^{2}+14x}{39}=\frac{9}{39}$$

Mā te whakawehe ki te 39 ka wetekia te whakareanga ki te 39.

$$x^{2}+\frac{14}{39}x=\frac{9}{39}$$

Whakahekea te hautanga \frac{9}{39} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 3.

$$x^{2}+\frac{14}{39}x=\frac{3}{13}$$

Whakawehea te \frac{14}{39}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{7}{39}. Nā, tāpiria te pūrua o te \frac{7}{39} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{14}{39}x+\left(\frac{7}{39}\right)^{2}=\frac{3}{13}+\left(\frac{7}{39}\right)^{2}$$

Pūruatia \frac{7}{39} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{14}{39}x+\frac{49}{1521}=\frac{3}{13}+\frac{49}{1521}$$

Tāpiri \frac{3}{13} ki te \frac{49}{1521} 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.

$$x^{2}+\frac{14}{39}x+\frac{49}{1521}=\frac{400}{1521}$$

Tauwehea x^{2}+\frac{14}{39}x+\frac{49}{1521}. 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}.

$$\left(x+\frac{7}{39}\right)^{2}=\frac{400}{1521}$$

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

$$\sqrt{\left(x+\frac{7}{39}\right)^{2}}=\sqrt{\frac{400}{1521}}$$

Whakarūnātia.

$$x+\frac{7}{39}=\frac{20}{39} x+\frac{7}{39}=-\frac{20}{39}$$

Me tango \frac{7}{39} mai i ngā taha e rua o te whārite.

$$x=\frac{1}{3} x=-\frac{9}{13}$$