Whakaoti i te 10x^2+7x-12=0

$$10 x ^ { 2 } + 7 x - 12 = 0$$

Whakaoti mō x

$$x = -\frac{3}{2} = -1\frac{1}{2} = -1.5$$

$$x=\frac{4}{5}=0.8$$

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 10x^{2}+ax+bx-12. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=7 ab=10\left(-12\right)=-120$$

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 10x^{2}+7x-12 hei \left(10x^{2}-8x\right)+\left(15x-12\right).

$$\left(10x^{2}-8x\right)+\left(15x-12\right)$$

Tauwehea te 2x i te tuatahi me te 3 i te rōpū tuarua.

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

Whakatauwehea atu te kīanga pātahi 5x-4 mā te whakamahi i te āhuatanga tātai tohatoha.

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

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

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

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.

$$10x^{2}+7x-12=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 10 mō a, 7 mō b, me -12 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-7±\sqrt{7^{2}-4\times 10\left(-12\right)}}{2\times 10}$$

Pūrua 7.

$$x=\frac{-7±\sqrt{49-4\times 10\left(-12\right)}}{2\times 10}$$

Whakareatia -4 ki te 10.

$$x=\frac{-7±\sqrt{49-40\left(-12\right)}}{2\times 10}$$

Whakareatia -40 ki te -12.

$$x=\frac{-7±\sqrt{49+480}}{2\times 10}$$

Tāpiri 49 ki te 480.

$$x=\frac{-7±\sqrt{529}}{2\times 10}$$

Tuhia te pūtakerua o te 529.

$$x=\frac{-7±23}{2\times 10}$$

Whakareatia 2 ki te 10.

$$x=\frac{-7±23}{20}$$

Nā, me whakaoti te whārite x=\frac{-7±23}{20} ina he tāpiri te ±. Tāpiri -7 ki te 23.

$$x=\frac{16}{20}$$

Whakahekea te hautanga \frac{16}{20} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

$$x=\frac{4}{5}$$

Nā, me whakaoti te whārite x=\frac{-7±23}{20} ina he tango te ±. Tango 23 mai i -7.

$$x=-\frac{30}{20}$$

Whakahekea te hautanga \frac{-30}{20} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 10.

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

Kua oti te whārite te whakatau.

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

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.

$$10x^{2}+7x-12=0$$

Me tāpiri 12 ki ngā taha e rua o te whārite.

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

Mā te tango i te -12 i a ia ake anō ka toe ko te 0.

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

Tango -12 mai i 0.

$$10x^{2}+7x=12$$

Whakawehea ngā taha e rua ki te 10.

$$\frac{10x^{2}+7x}{10}=\frac{12}{10}$$

Mā te whakawehe ki te 10 ka wetekia te whakareanga ki te 10.

$$x^{2}+\frac{7}{10}x=\frac{12}{10}$$

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

$$x^{2}+\frac{7}{10}x=\frac{6}{5}$$

Whakawehea te \frac{7}{10}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{7}{20}. Nā, tāpiria te pūrua o te \frac{7}{20} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{7}{10}x+\left(\frac{7}{20}\right)^{2}=\frac{6}{5}+\left(\frac{7}{20}\right)^{2}$$

Pūruatia \frac{7}{20} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{7}{10}x+\frac{49}{400}=\frac{6}{5}+\frac{49}{400}$$

Tāpiri \frac{6}{5} ki te \frac{49}{400} 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.

$$x^{2}+\frac{7}{10}x+\frac{49}{400}=\frac{529}{400}$$

Tauwehea x^{2}+\frac{7}{10}x+\frac{49}{400}. 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}.

$$\left(x+\frac{7}{20}\right)^{2}=\frac{529}{400}$$

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

$$\sqrt{\left(x+\frac{7}{20}\right)^{2}}=\sqrt{\frac{529}{400}}$$

Whakarūnātia.

$$x+\frac{7}{20}=\frac{23}{20} x+\frac{7}{20}=-\frac{23}{20}$$

Me tango \frac{7}{20} mai i ngā taha e rua o te whārite.

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