Whakaoti i te 4a^2-5a+1=0

$$4 { a }^{ 2 } -5a+1 = 0$$

Whakaoti mō a

$$a=\frac{1}{4}=0.25$$

$$a=1$$

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 4a^{2}+aa+ba+1. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-5 ab=4\times 1=4$$

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 4.

$$-1,-4 -2,-2$$

Tātaihia te tapeke mō ia takirua.

$$-1-4=-5 -2-2=-4$$

Ko te otinga te takirua ka hoatu i te tapeke -5.

$$a=-4 b=-1$$

Tuhia anō te 4a^{2}-5a+1 hei \left(4a^{2}-4a\right)+\left(-a+1\right).

$$\left(4a^{2}-4a\right)+\left(-a+1\right)$$

Tauwehea te 4a i te tuatahi me te -1 i te rōpū tuarua.

$$4a\left(a-1\right)-\left(a-1\right)$$

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

$$\left(a-1\right)\left(4a-1\right)$$

Hei kimi otinga whārite, me whakaoti te a-1=0 me te 4a-1=0.

$$a=1 a=\frac{1}{4}$$

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.

$$4a^{2}-5a+1=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 4 mō a, -5 mō b, me 1 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Pūrua -5.

$$a=\frac{-\left(-5\right)±\sqrt{25-4\times 4}}{2\times 4}$$

Whakareatia -4 ki te 4.

$$a=\frac{-\left(-5\right)±\sqrt{25-16}}{2\times 4}$$

Tāpiri 25 ki te -16.

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

Tuhia te pūtakerua o te 9.

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

Ko te tauaro o -5 ko 5.

$$a=\frac{5±3}{2\times 4}$$

Whakareatia 2 ki te 4.

$$a=\frac{5±3}{8}$$

Nā, me whakaoti te whārite a=\frac{5±3}{8} ina he tāpiri te ±. Tāpiri 5 ki te 3.

$$a=\frac{8}{8}$$

Whakawehe 8 ki te 8.

$$a=1$$

Nā, me whakaoti te whārite a=\frac{5±3}{8} ina he tango te ±. Tango 3 mai i 5.

$$a=\frac{2}{8}$$

Whakahekea te hautanga \frac{2}{8} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$a=\frac{1}{4}$$

Kua oti te whārite te whakatau.

$$a=1 a=\frac{1}{4}$$

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.

$$4a^{2}-5a+1=0$$

Me tango 1 mai i ngā taha e rua o te whārite.

$$4a^{2}-5a+1-1=-1$$

Mā te tango i te 1 i a ia ake anō ka toe ko te 0.

$$4a^{2}-5a=-1$$

Whakawehea ngā taha e rua ki te 4.

$$\frac{4a^{2}-5a}{4}=-\frac{1}{4}$$

Mā te whakawehe ki te 4 ka wetekia te whakareanga ki te 4.

$$a^{2}-\frac{5}{4}a=-\frac{1}{4}$$

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

$$a^{2}-\frac{5}{4}a+\left(-\frac{5}{8}\right)^{2}=-\frac{1}{4}+\left(-\frac{5}{8}\right)^{2}$$

Pūruatia -\frac{5}{8} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$a^{2}-\frac{5}{4}a+\frac{25}{64}=-\frac{1}{4}+\frac{25}{64}$$

Tāpiri -\frac{1}{4} ki te \frac{25}{64} 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.

$$a^{2}-\frac{5}{4}a+\frac{25}{64}=\frac{9}{64}$$

Tauwehea a^{2}-\frac{5}{4}a+\frac{25}{64}. 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}.

$$\left(a-\frac{5}{8}\right)^{2}=\frac{9}{64}$$

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

$$\sqrt{\left(a-\frac{5}{8}\right)^{2}}=\sqrt{\frac{9}{64}}$$

Whakarūnātia.

$$a-\frac{5}{8}=\frac{3}{8} a-\frac{5}{8}=-\frac{3}{8}$$

Me tāpiri \frac{5}{8} ki ngā taha e rua o te whārite.

$$a=1 a=\frac{1}{4}$$