Whakaoti i te 4u^2+25=-20u

$$4 u ^ { 2 } + 25 = - 20 u$$

Whakaoti mō u

$$u = -\frac{5}{2} = -2\frac{1}{2} = -2.5$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Me tāpiri te 20u ki ngā taha e rua.

$$4u^{2}+25+20u=0$$

Hurinahatia te pūrau ki te āhua tānga ngahuru. Whakaraupapahia ngā kīanga tau mai i te pū teitei rawa ki te mea iti rawa.

$$4u^{2}+20u+25=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 4u^{2}+au+bu+25. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=20 ab=4\times 25=100$$

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 100.

$$1,100 2,50 4,25 5,20 10,10$$

Tātaihia te tapeke mō ia takirua.

$$1+100=101 2+50=52 4+25=29 5+20=25 10+10=20$$

Ko te otinga te takirua ka hoatu i te tapeke 20.

$$a=10 b=10$$

Tuhia anō te 4u^{2}+20u+25 hei \left(4u^{2}+10u\right)+\left(10u+25\right).

$$\left(4u^{2}+10u\right)+\left(10u+25\right)$$

Tauwehea te 2u i te tuatahi me te 5 i te rōpū tuarua.

$$2u\left(2u+5\right)+5\left(2u+5\right)$$

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

$$\left(2u+5\right)\left(2u+5\right)$$

Tuhia anōtia hei pūrua huarua.

$$\left(2u+5\right)^{2}$$

Hei kimi i te otinga whārite, whakaotia te 2u+5=0.

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

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Me tāpiri te 20u ki ngā taha e rua.

$$4u^{2}+25+20u=0$$

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.

$$4u^{2}+20u+25=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 4 mō a, 20 mō b, me 25 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$u=\frac{-20±\sqrt{20^{2}-4\times 4\times 25}}{2\times 4}$$

Pūrua 20.

$$u=\frac{-20±\sqrt{400-4\times 4\times 25}}{2\times 4}$$

Whakareatia -4 ki te 4.

$$u=\frac{-20±\sqrt{400-16\times 25}}{2\times 4}$$

Whakareatia -16 ki te 25.

$$u=\frac{-20±\sqrt{400-400}}{2\times 4}$$

Tāpiri 400 ki te -400.

$$u=\frac{-20±\sqrt{0}}{2\times 4}$$

Tuhia te pūtakerua o te 0.

$$u=-\frac{20}{2\times 4}$$

Whakareatia 2 ki te 4.

$$u=-\frac{20}{8}$$

Whakahekea te hautanga \frac{-20}{8} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

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

Ngā Upane mō te Whakaoti i te Pūrua

Me tāpiri te 20u ki ngā taha e rua.

$$4u^{2}+25+20u=0$$

Tangohia te 25 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$4u^{2}+20u=-25$$

Whakawehea ngā taha e rua ki te 4.

$$\frac{4u^{2}+20u}{4}=-\frac{25}{4}$$

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

$$u^{2}+\frac{20}{4}u=-\frac{25}{4}$$

Whakawehe 20 ki te 4.

$$u^{2}+5u=-\frac{25}{4}$$

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

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

Pūruatia \frac{5}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$u^{2}+5u+\frac{25}{4}=\frac{-25+25}{4}$$

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

$$u^{2}+5u+\frac{25}{4}=0$$

Tauwehea u^{2}+5u+\frac{25}{4}. 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}.

$$\left(u+\frac{5}{2}\right)^{2}=0$$

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

$$\sqrt{\left(u+\frac{5}{2}\right)^{2}}=\sqrt{0}$$

Whakarūnātia.

$$u+\frac{5}{2}=0 u+\frac{5}{2}=0$$

Me tango \frac{5}{2} mai i ngā taha e rua o te whārite.

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

Kua oti te whārite te whakatau. He ōrite ngā whakatau.

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