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

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

Whakaoti mō x

$$x=-1$$

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

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 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 -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.

$$a=-2 b=5$$

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

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

Whakatauwehea atu x i te 5x^{2}-2x.

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

Whakatauwehea atu te kīanga pātahi 5x-2 mā te whakamahi i te āhuatanga tātai tohatoha.

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

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

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

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}.

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

Pūrua 3.

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

Whakareatia -4 ki te 5.

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

Whakareatia -20 ki te -2.

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

Tāpiri 9 ki te 40.

$$x=\frac{-3±\sqrt{49}}{2\times 5}$$

Tuhia te pūtakerua o te 49.

$$x=\frac{-3±7}{2\times 5}$$

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.

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

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

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

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

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

Whakawehe -10 ki te 10.

$$x=-1$$

Kua oti te whārite te whakatau.

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

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

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

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 tango \frac{3}{10} mai i ngā taha e rua o te whārite.

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