Whakaoti i te 2r^2+5r+2=0

$$2 r ^ { 2 } + 5 r + 2 = 0$$

Whakaoti mō r

$$r=-2$$

$$r=-\frac{1}{2}=-0.5$$

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 2r^{2}+ar+br+2. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=5 ab=2\times 2=4$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōrunga te a+b, he tōrunga 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.

$$a=1 b=4$$

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

$$\left(2r^{2}+r\right)+\left(4r+2\right)$$

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

$$r\left(2r+1\right)+2\left(2r+1\right)$$

Whakatauwehea atu te kīanga pātahi 2r+1 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(2r+1\right)\left(r+2\right)$$

Hei kimi otinga whārite, me whakaoti te 2r+1=0 me te r+2=0.

$$r=-\frac{1}{2} r=-2$$

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.

$$2r^{2}+5r+2=0$$

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

$$r=\frac{-5±\sqrt{5^{2}-4\times 2\times 2}}{2\times 2}$$

Pūrua 5.

$$r=\frac{-5±\sqrt{25-4\times 2\times 2}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$r=\frac{-5±\sqrt{25-8\times 2}}{2\times 2}$$

Whakareatia -8 ki te 2.

$$r=\frac{-5±\sqrt{25-16}}{2\times 2}$$

Tāpiri 25 ki te -16.

$$r=\frac{-5±\sqrt{9}}{2\times 2}$$

Tuhia te pūtakerua o te 9.

$$r=\frac{-5±3}{2\times 2}$$

Whakareatia 2 ki te 2.

$$r=\frac{-5±3}{4}$$

Nā, me whakaoti te whārite r=\frac{-5±3}{4} ina he tāpiri te ±. Tāpiri -5 ki te 3.

$$r=-\frac{2}{4}$$

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

$$r=-\frac{1}{2}$$

Nā, me whakaoti te whārite r=\frac{-5±3}{4} ina he tango te ±. Tango 3 mai i -5.

$$r=-\frac{8}{4}$$

Whakawehe -8 ki te 4.

$$r=-2$$

Kua oti te whārite te whakatau.

$$r=-\frac{1}{2} r=-2$$

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.

$$2r^{2}+5r+2=0$$

Me tango 2 mai i ngā taha e rua o te whārite.

$$2r^{2}+5r+2-2=-2$$

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

$$2r^{2}+5r=-2$$

Whakawehea ngā taha e rua ki te 2.

$$\frac{2r^{2}+5r}{2}=-\frac{2}{2}$$

Mā te whakawehe ki te 2 ka wetekia te whakareanga ki te 2.

$$r^{2}+\frac{5}{2}r=-\frac{2}{2}$$

Whakawehe -2 ki te 2.

$$r^{2}+\frac{5}{2}r=-1$$

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

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

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

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

Tāpiri -1 ki te \frac{25}{16}.

$$r^{2}+\frac{5}{2}r+\frac{25}{16}=\frac{9}{16}$$

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

$$\left(r+\frac{5}{4}\right)^{2}=\frac{9}{16}$$

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

$$\sqrt{\left(r+\frac{5}{4}\right)^{2}}=\sqrt{\frac{9}{16}}$$

Whakarūnātia.

$$r+\frac{5}{4}=\frac{3}{4} r+\frac{5}{4}=-\frac{3}{4}$$

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

$$r=-\frac{1}{2} r=-2$$