Whakaoti i te 20p^2+33p+16=6

$$20 p ^ { 2 } + 33 p + 16 = 6$$

Whakaoti mō p

$$p = -\frac{5}{4} = -1\frac{1}{4} = -1.25$$

$$p=-\frac{2}{5}=-0.4$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tangohia te 6 mai i ngā taha e rua.

$$20p^{2}+33p+16-6=0$$

Tangohia te 6 i te 16, ka 10.

$$20p^{2}+33p+10=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 20p^{2}+ap+bp+10. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=33 ab=20\times 10=200$$

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 200.

$$1,200 2,100 4,50 5,40 8,25 10,20$$

Tātaihia te tapeke mō ia takirua.

$$1+200=201 2+100=102 4+50=54 5+40=45 8+25=33 10+20=30$$

Ko te otinga te takirua ka hoatu i te tapeke 33.

$$a=8 b=25$$

Tuhia anō te 20p^{2}+33p+10 hei \left(20p^{2}+8p\right)+\left(25p+10\right).

$$\left(20p^{2}+8p\right)+\left(25p+10\right)$$

Tauwehea te 4p i te tuatahi me te 5 i te rōpū tuarua.

$$4p\left(5p+2\right)+5\left(5p+2\right)$$

Whakatauwehea atu te kīanga pātahi 5p+2 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(5p+2\right)\left(4p+5\right)$$

Hei kimi otinga whārite, me whakaoti te 5p+2=0 me te 4p+5=0.

$$p=-\frac{2}{5} p=-\frac{5}{4}$$

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.

$$20p^{2}+33p+16=6$$

Me tango 6 mai i ngā taha e rua o te whārite.

$$20p^{2}+33p+16-6=6-6$$

Mā te tango i te 6 i a ia ake anō ka toe ko te 0.

$$20p^{2}+33p+16-6=0$$

Tango 6 mai i 16.

$$20p^{2}+33p+10=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 20 mō a, 33 mō b, me 10 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$p=\frac{-33±\sqrt{33^{2}-4\times 20\times 10}}{2\times 20}$$

Pūrua 33.

$$p=\frac{-33±\sqrt{1089-4\times 20\times 10}}{2\times 20}$$

Whakareatia -4 ki te 20.

$$p=\frac{-33±\sqrt{1089-80\times 10}}{2\times 20}$$

Whakareatia -80 ki te 10.

$$p=\frac{-33±\sqrt{1089-800}}{2\times 20}$$

Tāpiri 1089 ki te -800.

$$p=\frac{-33±\sqrt{289}}{2\times 20}$$

Tuhia te pūtakerua o te 289.

$$p=\frac{-33±17}{2\times 20}$$

Whakareatia 2 ki te 20.

$$p=\frac{-33±17}{40}$$

Nā, me whakaoti te whārite p=\frac{-33±17}{40} ina he tāpiri te ±. Tāpiri -33 ki te 17.

$$p=-\frac{16}{40}$$

Whakahekea te hautanga \frac{-16}{40} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 8.

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

Nā, me whakaoti te whārite p=\frac{-33±17}{40} ina he tango te ±. Tango 17 mai i -33.

$$p=-\frac{50}{40}$$

Whakahekea te hautanga \frac{-50}{40} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 10.

$$p=-\frac{5}{4}$$

Kua oti te whārite te whakatau.

$$p=-\frac{2}{5} p=-\frac{5}{4}$$

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.

$$20p^{2}+33p+16=6$$

Me tango 16 mai i ngā taha e rua o te whārite.

$$20p^{2}+33p+16-16=6-16$$

Mā te tango i te 16 i a ia ake anō ka toe ko te 0.

$$20p^{2}+33p=6-16$$

Tango 16 mai i 6.

$$20p^{2}+33p=-10$$

Whakawehea ngā taha e rua ki te 20.

$$\frac{20p^{2}+33p}{20}=-\frac{10}{20}$$

Mā te whakawehe ki te 20 ka wetekia te whakareanga ki te 20.

$$p^{2}+\frac{33}{20}p=-\frac{10}{20}$$

Whakahekea te hautanga \frac{-10}{20} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 10.

$$p^{2}+\frac{33}{20}p=-\frac{1}{2}$$

Whakawehea te \frac{33}{20}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{33}{40}. Nā, tāpiria te pūrua o te \frac{33}{40} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$p^{2}+\frac{33}{20}p+\left(\frac{33}{40}\right)^{2}=-\frac{1}{2}+\left(\frac{33}{40}\right)^{2}$$

Pūruatia \frac{33}{40} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$p^{2}+\frac{33}{20}p+\frac{1089}{1600}=-\frac{1}{2}+\frac{1089}{1600}$$

Tāpiri -\frac{1}{2} ki te \frac{1089}{1600} 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.

$$p^{2}+\frac{33}{20}p+\frac{1089}{1600}=\frac{289}{1600}$$

Tauwehea p^{2}+\frac{33}{20}p+\frac{1089}{1600}. 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}.

$$\left(p+\frac{33}{40}\right)^{2}=\frac{289}{1600}$$

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

$$\sqrt{\left(p+\frac{33}{40}\right)^{2}}=\sqrt{\frac{289}{1600}}$$

Whakarūnātia.

$$p+\frac{33}{40}=\frac{17}{40} p+\frac{33}{40}=-\frac{17}{40}$$

Me tango \frac{33}{40} mai i ngā taha e rua o te whārite.

$$p=-\frac{2}{5} p=-\frac{5}{4}$$