Whakaoti i te (2y+3)^2+y^2=4

$$( 2 y + 3 ) ^ { 2 } + y ^ { 2 } = 4$$

Whakaoti mō y

$$y=\frac{\sqrt{11}-6}{5}\approx -0.536675042$$

$$y=\frac{-\sqrt{11}-6}{5}\approx -1.863324958$$

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(2y+3\right)^{2}.

$$4y^{2}+12y+9+y^{2}=4$$

Pahekotia te 4y^{2} me y^{2}, ka 5y^{2}.

$$5y^{2}+12y+9=4$$

Tangohia te 4 mai i ngā taha e rua.

$$5y^{2}+12y+9-4=0$$

Tangohia te 4 i te 9, ka 5.

$$5y^{2}+12y+5=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 5 mō a, 12 mō b, me 5 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-12±\sqrt{12^{2}-4\times 5\times 5}}{2\times 5}$$

Pūrua 12.

$$y=\frac{-12±\sqrt{144-4\times 5\times 5}}{2\times 5}$$

Whakareatia -4 ki te 5.

$$y=\frac{-12±\sqrt{144-20\times 5}}{2\times 5}$$

Whakareatia -20 ki te 5.

$$y=\frac{-12±\sqrt{144-100}}{2\times 5}$$

Tāpiri 144 ki te -100.

$$y=\frac{-12±\sqrt{44}}{2\times 5}$$

Tuhia te pūtakerua o te 44.

$$y=\frac{-12±2\sqrt{11}}{2\times 5}$$

Whakareatia 2 ki te 5.

$$y=\frac{-12±2\sqrt{11}}{10}$$

Nā, me whakaoti te whārite y=\frac{-12±2\sqrt{11}}{10} ina he tāpiri te ±. Tāpiri -12 ki te 2\sqrt{11}.

$$y=\frac{2\sqrt{11}-12}{10}$$

Whakawehe -12+2\sqrt{11} ki te 10.

$$y=\frac{\sqrt{11}-6}{5}$$

Nā, me whakaoti te whārite y=\frac{-12±2\sqrt{11}}{10} ina he tango te ±. Tango 2\sqrt{11} mai i -12.

$$y=\frac{-2\sqrt{11}-12}{10}$$

Whakawehe -12-2\sqrt{11} ki te 10.

$$y=\frac{-\sqrt{11}-6}{5}$$

Kua oti te whārite te whakatau.

$$y=\frac{\sqrt{11}-6}{5} y=\frac{-\sqrt{11}-6}{5}$$

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(2y+3\right)^{2}.

$$4y^{2}+12y+9+y^{2}=4$$

Pahekotia te 4y^{2} me y^{2}, ka 5y^{2}.

$$5y^{2}+12y+9=4$$

Tangohia te 9 mai i ngā taha e rua.

$$5y^{2}+12y=4-9$$

Tangohia te 9 i te 4, ka -5.

$$5y^{2}+12y=-5$$

Whakawehea ngā taha e rua ki te 5.

$$\frac{5y^{2}+12y}{5}=-\frac{5}{5}$$

Mā te whakawehe ki te 5 ka wetekia te whakareanga ki te 5.

$$y^{2}+\frac{12}{5}y=-\frac{5}{5}$$

Whakawehe -5 ki te 5.

$$y^{2}+\frac{12}{5}y=-1$$

Whakawehea te \frac{12}{5}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{6}{5}. Nā, tāpiria te pūrua o te \frac{6}{5} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$y^{2}+\frac{12}{5}y+\left(\frac{6}{5}\right)^{2}=-1+\left(\frac{6}{5}\right)^{2}$$

Pūruatia \frac{6}{5} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$y^{2}+\frac{12}{5}y+\frac{36}{25}=-1+\frac{36}{25}$$

Tāpiri -1 ki te \frac{36}{25}.

$$y^{2}+\frac{12}{5}y+\frac{36}{25}=\frac{11}{25}$$

Tauwehea y^{2}+\frac{12}{5}y+\frac{36}{25}. 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}.

$$\left(y+\frac{6}{5}\right)^{2}=\frac{11}{25}$$

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

$$\sqrt{\left(y+\frac{6}{5}\right)^{2}}=\sqrt{\frac{11}{25}}$$

Whakarūnātia.

$$y+\frac{6}{5}=\frac{\sqrt{11}}{5} y+\frac{6}{5}=-\frac{\sqrt{11}}{5}$$

Me tango \frac{6}{5} mai i ngā taha e rua o te whārite.

$$y=\frac{\sqrt{11}-6}{5} y=\frac{-\sqrt{11}-6}{5}$$