Whakaoti i te 5x^2+8x-4=0

$$5 x ^ { 2 } + 8 x - 4 = 0$$

Whakaoti mō x

$$x=-2$$

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

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-4. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=8 ab=5\left(-4\right)=-20$$

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 -20.

$$-1,20 -2,10 -4,5$$

Tātaihia te tapeke mō ia takirua.

$$-1+20=19 -2+10=8 -4+5=1$$

Ko te otinga te takirua ka hoatu i te tapeke 8.

$$a=-2 b=10$$

Tuhia anō te 5x^{2}+8x-4 hei \left(5x^{2}-2x\right)+\left(10x-4\right).

$$\left(5x^{2}-2x\right)+\left(10x-4\right)$$

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

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

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

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

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

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

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.

$$5x^{2}+8x-4=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 5 mō a, 8 mō b, me -4 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-8±\sqrt{8^{2}-4\times 5\left(-4\right)}}{2\times 5}$$

Pūrua 8.

$$x=\frac{-8±\sqrt{64-4\times 5\left(-4\right)}}{2\times 5}$$

Whakareatia -4 ki te 5.

$$x=\frac{-8±\sqrt{64-20\left(-4\right)}}{2\times 5}$$

Whakareatia -20 ki te -4.

$$x=\frac{-8±\sqrt{64+80}}{2\times 5}$$

Tāpiri 64 ki te 80.

$$x=\frac{-8±\sqrt{144}}{2\times 5}$$

Tuhia te pūtakerua o te 144.

$$x=\frac{-8±12}{2\times 5}$$

Whakareatia 2 ki te 5.

$$x=\frac{-8±12}{10}$$

Nā, me whakaoti te whārite x=\frac{-8±12}{10} ina he tāpiri te ±. Tāpiri -8 ki te 12.

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

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{-8±12}{10} ina he tango te ±. Tango 12 mai i -8.

$$x=-\frac{20}{10}$$

Whakawehe -20 ki te 10.

$$x=-2$$

Kua oti te whārite te whakatau.

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

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.

$$5x^{2}+8x-4=0$$

Me tāpiri 4 ki ngā taha e rua o te whārite.

$$5x^{2}+8x-4-\left(-4\right)=-\left(-4\right)$$

Mā te tango i te -4 i a ia ake anō ka toe ko te 0.

$$5x^{2}+8x=-\left(-4\right)$$

Tango -4 mai i 0.

$$5x^{2}+8x=4$$

Whakawehea ngā taha e rua ki te 5.

$$\frac{5x^{2}+8x}{5}=\frac{4}{5}$$

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

$$x^{2}+\frac{8}{5}x=\frac{4}{5}$$

Whakawehea te \frac{8}{5}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{4}{5}. Nā, tāpiria te pūrua o te \frac{4}{5} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{8}{5}x+\left(\frac{4}{5}\right)^{2}=\frac{4}{5}+\left(\frac{4}{5}\right)^{2}$$

Pūruatia \frac{4}{5} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{8}{5}x+\frac{16}{25}=\frac{4}{5}+\frac{16}{25}$$

Tāpiri \frac{4}{5} ki te \frac{16}{25} 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.

$$x^{2}+\frac{8}{5}x+\frac{16}{25}=\frac{36}{25}$$

Tauwehea x^{2}+\frac{8}{5}x+\frac{16}{25}. 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}.

$$\left(x+\frac{4}{5}\right)^{2}=\frac{36}{25}$$

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

$$\sqrt{\left(x+\frac{4}{5}\right)^{2}}=\sqrt{\frac{36}{25}}$$

Whakarūnātia.

$$x+\frac{4}{5}=\frac{6}{5} x+\frac{4}{5}=-\frac{6}{5}$$

Me tango \frac{4}{5} mai i ngā taha e rua o te whārite.

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