Whakaoti i te 5x^2-8x+3=0

$$5 x ^ { 2 } - 8 x + 3 = 0$$

Whakaoti mō x

$$x=1$$

$$x=\frac{3}{5}=0.6$$

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+3. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-8 ab=5\times 3=15$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōraro te a+b, he tōraro hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 15.

$$-1,-15 -3,-5$$

Tātaihia te tapeke mō ia takirua.

$$-1-15=-16 -3-5=-8$$

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

$$a=-5 b=-3$$

Tuhia anō te 5x^{2}-8x+3 hei \left(5x^{2}-5x\right)+\left(-3x+3\right).

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

Tauwehea te 5x i te tuatahi me te -3 i te rōpū tuarua.

$$5x\left(x-1\right)-3\left(x-1\right)$$

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

$$\left(x-1\right)\left(5x-3\right)$$

Hei kimi otinga whārite, me whakaoti te x-1=0 me te 5x-3=0.

$$x=1 x=\frac{3}{5}$$

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.

$$5x^{2}-8x+3=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 5 mō a, -8 mō b, me 3 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\times 5\times 3}}{2\times 5}$$

Pūrua -8.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\times 5\times 3}}{2\times 5}$$

Whakareatia -4 ki te 5.

$$x=\frac{-\left(-8\right)±\sqrt{64-20\times 3}}{2\times 5}$$

Whakareatia -20 ki te 3.

$$x=\frac{-\left(-8\right)±\sqrt{64-60}}{2\times 5}$$

Tāpiri 64 ki te -60.

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

Tuhia te pūtakerua o te 4.

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

Ko te tauaro o -8 ko 8.

$$x=\frac{8±2}{2\times 5}$$

Whakareatia 2 ki te 5.

$$x=\frac{8±2}{10}$$

Nā, me whakaoti te whārite x=\frac{8±2}{10} ina he tāpiri te ±. Tāpiri 8 ki te 2.

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

Whakawehe 10 ki te 10.

$$x=1$$

Nā, me whakaoti te whārite x=\frac{8±2}{10} ina he tango te ±. Tango 2 mai i 8.

$$x=\frac{6}{10}$$

Whakahekea te hautanga \frac{6}{10} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x=\frac{3}{5}$$

Kua oti te whārite te whakatau.

$$x=1 x=\frac{3}{5}$$

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.

$$5x^{2}-8x+3=0$$

Me tango 3 mai i ngā taha e rua o te whārite.

$$5x^{2}-8x+3-3=-3$$

Mā te tango i te 3 i a ia ake anō ka toe ko te 0.

$$5x^{2}-8x=-3$$

Whakawehea ngā taha e rua ki te 5.

$$\frac{5x^{2}-8x}{5}=-\frac{3}{5}$$

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

$$x^{2}-\frac{8}{5}x=-\frac{3}{5}$$

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.

$$x^{2}-\frac{8}{5}x+\left(-\frac{4}{5}\right)^{2}=-\frac{3}{5}+\left(-\frac{4}{5}\right)^{2}$$

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

$$x^{2}-\frac{8}{5}x+\frac{16}{25}=-\frac{3}{5}+\frac{16}{25}$$

Tāpiri -\frac{3}{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.

$$x^{2}-\frac{8}{5}x+\frac{16}{25}=\frac{1}{25}$$

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

$$\left(x-\frac{4}{5}\right)^{2}=\frac{1}{25}$$

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

$$\sqrt{\left(x-\frac{4}{5}\right)^{2}}=\sqrt{\frac{1}{25}}$$

Whakarūnātia.

$$x-\frac{4}{5}=\frac{1}{5} x-\frac{4}{5}=-\frac{1}{5}$$

Me tāpiri \frac{4}{5} ki ngā taha e rua o te whārite.

$$x=1 x=\frac{3}{5}$$