Whakaoti i te left(u+1right)^2=2u^2+5u+3

$${ \left( u+1 \right) }^{ 2 } =2 { u }^{ 2 } +5u+3$$

Whakaoti mō u

$$u=-1$$

$$u=-2$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(u+1\right)^{2}.

$$u^{2}+2u+1=2u^{2}+5u+3$$

Tangohia te 2u^{2} mai i ngā taha e rua.

$$u^{2}+2u+1-2u^{2}=5u+3$$

Pahekotia te u^{2} me -2u^{2}, ka -u^{2}.

$$-u^{2}+2u+1=5u+3$$

Tangohia te 5u mai i ngā taha e rua.

$$-u^{2}+2u+1-5u=3$$

Pahekotia te 2u me -5u, ka -3u.

$$-u^{2}-3u+1=3$$

Tangohia te 3 mai i ngā taha e rua.

$$-u^{2}-3u+1-3=0$$

Tangohia te 3 i te 1, ka -2.

$$-u^{2}-3u-2=0$$

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

$$a+b=-3 ab=-\left(-2\right)=2$$

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. Ko te takirua anake pērā ko te otinga pūnaha.

$$a=-1 b=-2$$

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

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

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

$$u\left(-u-1\right)+2\left(-u-1\right)$$

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

$$\left(-u-1\right)\left(u+2\right)$$

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

$$u=-1 u=-2$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(u+1\right)^{2}.

$$u^{2}+2u+1=2u^{2}+5u+3$$

Tangohia te 2u^{2} mai i ngā taha e rua.

$$u^{2}+2u+1-2u^{2}=5u+3$$

Pahekotia te u^{2} me -2u^{2}, ka -u^{2}.

$$-u^{2}+2u+1=5u+3$$

Tangohia te 5u mai i ngā taha e rua.

$$-u^{2}+2u+1-5u=3$$

Pahekotia te 2u me -5u, ka -3u.

$$-u^{2}-3u+1=3$$

Tangohia te 3 mai i ngā taha e rua.

$$-u^{2}-3u+1-3=0$$

Tangohia te 3 i te 1, ka -2.

$$-u^{2}-3u-2=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -1 mō a, -3 mō b, me -2 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$u=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\left(-1\right)\left(-2\right)}}{2\left(-1\right)}$$

Pūrua -3.

$$u=\frac{-\left(-3\right)±\sqrt{9-4\left(-1\right)\left(-2\right)}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$u=\frac{-\left(-3\right)±\sqrt{9+4\left(-2\right)}}{2\left(-1\right)}$$

Whakareatia 4 ki te -2.

$$u=\frac{-\left(-3\right)±\sqrt{9-8}}{2\left(-1\right)}$$

Tāpiri 9 ki te -8.

$$u=\frac{-\left(-3\right)±\sqrt{1}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te 1.

$$u=\frac{-\left(-3\right)±1}{2\left(-1\right)}$$

Ko te tauaro o -3 ko 3.

$$u=\frac{3±1}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$u=\frac{3±1}{-2}$$

Nā, me whakaoti te whārite u=\frac{3±1}{-2} ina he tāpiri te ±. Tāpiri 3 ki te 1.

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

Whakawehe 4 ki te -2.

$$u=-2$$

Nā, me whakaoti te whārite u=\frac{3±1}{-2} ina he tango te ±. Tango 1 mai i 3.

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

Whakawehe 2 ki te -2.

$$u=-1$$

Kua oti te whārite te whakatau.

$$u=-2 u=-1$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(u+1\right)^{2}.

$$u^{2}+2u+1=2u^{2}+5u+3$$

Tangohia te 2u^{2} mai i ngā taha e rua.

$$u^{2}+2u+1-2u^{2}=5u+3$$

Pahekotia te u^{2} me -2u^{2}, ka -u^{2}.

$$-u^{2}+2u+1=5u+3$$

Tangohia te 5u mai i ngā taha e rua.

$$-u^{2}+2u+1-5u=3$$

Pahekotia te 2u me -5u, ka -3u.

$$-u^{2}-3u+1=3$$

Tangohia te 1 mai i ngā taha e rua.

$$-u^{2}-3u=3-1$$

Tangohia te 1 i te 3, ka 2.

$$-u^{2}-3u=2$$

Whakawehea ngā taha e rua ki te -1.

$$\frac{-u^{2}-3u}{-1}=\frac{2}{-1}$$

Mā te whakawehe ki te -1 ka wetekia te whakareanga ki te -1.

$$u^{2}+\left(-\frac{3}{-1}\right)u=\frac{2}{-1}$$

Whakawehe -3 ki te -1.

$$u^{2}+3u=\frac{2}{-1}$$

Whakawehe 2 ki te -1.

$$u^{2}+3u=-2$$

Whakawehea te 3, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{3}{2}. Nā, tāpiria te pūrua o te \frac{3}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$u^{2}+3u+\left(\frac{3}{2}\right)^{2}=-2+\left(\frac{3}{2}\right)^{2}$$

Pūruatia \frac{3}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$u^{2}+3u+\frac{9}{4}=-2+\frac{9}{4}$$

Tāpiri -2 ki te \frac{9}{4}.

$$u^{2}+3u+\frac{9}{4}=\frac{1}{4}$$

Tauwehea te u^{2}+3u+\frac{9}{4}. 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}.

$$\left(u+\frac{3}{2}\right)^{2}=\frac{1}{4}$$

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

$$\sqrt{\left(u+\frac{3}{2}\right)^{2}}=\sqrt{\frac{1}{4}}$$

Whakarūnātia.

$$u+\frac{3}{2}=\frac{1}{2} u+\frac{3}{2}=-\frac{1}{2}$$

Me tango \frac{3}{2} mai i ngā taha e rua o te whārite.

$$u=-1 u=-2$$