Whakaoti i te 5x^2-3x-2=0

$$5 x ^ { 2 } - 3 x - 2 = 0$$

Whakaoti mō x

$$x=-\frac{2}{5}=-0.4$$

$$x=1$$

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 5x^{2}+ax+bx-2. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-3 ab=5\left(-2\right)=-10$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōraro te a+b, he nui ake te uara pū o te tau tōraro i tō te tōrunga. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -10.

$$1,-10 2,-5$$

Tātaihia te tapeke mō ia takirua.

$$1-10=-9 2-5=-3$$

Ko te otinga te takirua ka hoatu i te tapeke -3.

$$a=-5 b=2$$

Tuhia anō te 5x^{2}-3x-2 hei \left(5x^{2}-5x\right)+\left(2x-2\right).

$$\left(5x^{2}-5x\right)+\left(2x-2\right)$$

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

$$5x\left(x-1\right)+2\left(x-1\right)$$

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

$$\left(x-1\right)\left(5x+2\right)$$

Hei kimi otinga whārite, me whakaoti te x-1=0 me te 5x+2=0.

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

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.

$$5x^{2}-3x-2=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 5 mō a, -3 mō b, me -2 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\times 5\left(-2\right)}}{2\times 5}$$

Pūrua -3.

$$x=\frac{-\left(-3\right)±\sqrt{9-4\times 5\left(-2\right)}}{2\times 5}$$

Whakareatia -4 ki te 5.

$$x=\frac{-\left(-3\right)±\sqrt{9-20\left(-2\right)}}{2\times 5}$$

Whakareatia -20 ki te -2.

$$x=\frac{-\left(-3\right)±\sqrt{9+40}}{2\times 5}$$

Tāpiri 9 ki te 40.

$$x=\frac{-\left(-3\right)±\sqrt{49}}{2\times 5}$$

Tuhia te pūtakerua o te 49.

$$x=\frac{-\left(-3\right)±7}{2\times 5}$$

Ko te tauaro o -3 ko 3.

$$x=\frac{3±7}{2\times 5}$$

Whakareatia 2 ki te 5.

$$x=\frac{3±7}{10}$$

Nā, me whakaoti te whārite x=\frac{3±7}{10} ina he tāpiri te ±. Tāpiri 3 ki te 7.

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

Whakawehe 10 ki te 10.

$$x=1$$

Nā, me whakaoti te whārite x=\frac{3±7}{10} ina he tango te ±. Tango 7 mai i 3.

$$x=-\frac{4}{10}$$

Whakahekea te hautanga \frac{-4}{10} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

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

Kua oti te whārite te whakatau.

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

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.

$$5x^{2}-3x-2=0$$

Me tāpiri 2 ki ngā taha e rua o te whārite.

$$5x^{2}-3x-2-\left(-2\right)=-\left(-2\right)$$

Mā te tango i te -2 i a ia ake anō ka toe ko te 0.

$$5x^{2}-3x=-\left(-2\right)$$

Tango -2 mai i 0.

$$5x^{2}-3x=2$$

Whakawehea ngā taha e rua ki te 5.

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

Mā te whakawehe ki te 5 ka wetekia te whakareanga ki te 5.

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

Whakawehea te -\frac{3}{5}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{3}{10}. Nā, tāpiria te pūrua o te -\frac{3}{10} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{3}{5}x+\left(-\frac{3}{10}\right)^{2}=\frac{2}{5}+\left(-\frac{3}{10}\right)^{2}$$

Pūruatia -\frac{3}{10} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{3}{5}x+\frac{9}{100}=\frac{2}{5}+\frac{9}{100}$$

Tāpiri \frac{2}{5} ki te \frac{9}{100} 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.

$$x^{2}-\frac{3}{5}x+\frac{9}{100}=\frac{49}{100}$$

Tauwehea te x^{2}-\frac{3}{5}x+\frac{9}{100}. 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}.

$$\left(x-\frac{3}{10}\right)^{2}=\frac{49}{100}$$

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

$$\sqrt{\left(x-\frac{3}{10}\right)^{2}}=\sqrt{\frac{49}{100}}$$

Whakarūnātia.

$$x-\frac{3}{10}=\frac{7}{10} x-\frac{3}{10}=-\frac{7}{10}$$

Me tāpiri \frac{3}{10} ki ngā taha e rua o te whārite.

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