Whakaoti i te 6x^2+7x-20=0

$$6 x ^ { 2 } + 7 x - 20 = 0$$

Whakaoti mō x

$$x = -\frac{5}{2} = -2\frac{1}{2} = -2.5$$

$$x = \frac{4}{3} = 1\frac{1}{3} \approx 1.333333333$$

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 6x^{2}+ax+bx-20. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=7 ab=6\left(-20\right)=-120$$

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 -120.

$$-1,120 -2,60 -3,40 -4,30 -5,24 -6,20 -8,15 -10,12$$

Tātaihia te tapeke mō ia takirua.

$$-1+120=119 -2+60=58 -3+40=37 -4+30=26 -5+24=19 -6+20=14 -8+15=7 -10+12=2$$

Ko te otinga te takirua ka hoatu i te tapeke 7.

$$a=-8 b=15$$

Tuhia anō te 6x^{2}+7x-20 hei \left(6x^{2}-8x\right)+\left(15x-20\right).

$$\left(6x^{2}-8x\right)+\left(15x-20\right)$$

Tauwehea te 2x i te tuatahi me te 5 i te rōpū tuarua.

$$2x\left(3x-4\right)+5\left(3x-4\right)$$

Whakatauwehea atu te kīanga pātahi 3x-4 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(3x-4\right)\left(2x+5\right)$$

Hei kimi otinga whārite, me whakaoti te 3x-4=0 me te 2x+5=0.

$$x=\frac{4}{3} x=-\frac{5}{2}$$

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.

$$6x^{2}+7x-20=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 6 mō a, 7 mō b, me -20 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-7±\sqrt{7^{2}-4\times 6\left(-20\right)}}{2\times 6}$$

Pūrua 7.

$$x=\frac{-7±\sqrt{49-4\times 6\left(-20\right)}}{2\times 6}$$

Whakareatia -4 ki te 6.

$$x=\frac{-7±\sqrt{49-24\left(-20\right)}}{2\times 6}$$

Whakareatia -24 ki te -20.

$$x=\frac{-7±\sqrt{49+480}}{2\times 6}$$

Tāpiri 49 ki te 480.

$$x=\frac{-7±\sqrt{529}}{2\times 6}$$

Tuhia te pūtakerua o te 529.

$$x=\frac{-7±23}{2\times 6}$$

Whakareatia 2 ki te 6.

$$x=\frac{-7±23}{12}$$

Nā, me whakaoti te whārite x=\frac{-7±23}{12} ina he tāpiri te ±. Tāpiri -7 ki te 23.

$$x=\frac{16}{12}$$

Whakahekea te hautanga \frac{16}{12} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

$$x=\frac{4}{3}$$

Nā, me whakaoti te whārite x=\frac{-7±23}{12} ina he tango te ±. Tango 23 mai i -7.

$$x=-\frac{30}{12}$$

Whakahekea te hautanga \frac{-30}{12} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 6.

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

Kua oti te whārite te whakatau.

$$x=\frac{4}{3} x=-\frac{5}{2}$$

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.

$$6x^{2}+7x-20=0$$

Me tāpiri 20 ki ngā taha e rua o te whārite.

$$6x^{2}+7x-20-\left(-20\right)=-\left(-20\right)$$

Mā te tango i te -20 i a ia ake anō ka toe ko te 0.

$$6x^{2}+7x=-\left(-20\right)$$

Tango -20 mai i 0.

$$6x^{2}+7x=20$$

Whakawehea ngā taha e rua ki te 6.

$$\frac{6x^{2}+7x}{6}=\frac{20}{6}$$

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

$$x^{2}+\frac{7}{6}x=\frac{20}{6}$$

Whakahekea te hautanga \frac{20}{6} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x^{2}+\frac{7}{6}x=\frac{10}{3}$$

Whakawehea te \frac{7}{6}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{7}{12}. Nā, tāpiria te pūrua o te \frac{7}{12} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{7}{6}x+\left(\frac{7}{12}\right)^{2}=\frac{10}{3}+\left(\frac{7}{12}\right)^{2}$$

Pūruatia \frac{7}{12} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{7}{6}x+\frac{49}{144}=\frac{10}{3}+\frac{49}{144}$$

Tāpiri \frac{10}{3} ki te \frac{49}{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.

$$x^{2}+\frac{7}{6}x+\frac{49}{144}=\frac{529}{144}$$

Tauwehea x^{2}+\frac{7}{6}x+\frac{49}{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}.

$$\left(x+\frac{7}{12}\right)^{2}=\frac{529}{144}$$

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

$$\sqrt{\left(x+\frac{7}{12}\right)^{2}}=\sqrt{\frac{529}{144}}$$

Whakarūnātia.

$$x+\frac{7}{12}=\frac{23}{12} x+\frac{7}{12}=-\frac{23}{12}$$

Me tango \frac{7}{12} mai i ngā taha e rua o te whārite.

$$x=\frac{4}{3} x=-\frac{5}{2}$$