Whakaoti i te 1/2left(1-k/2right)left(2-kright)=left(k+2right)left(1-k/2right)

$$\frac{ 1 }{ 2 } \left( 1- \frac{ k }{ 2 } \right) \left( 2-k \right) = \left( k+2 \right) \left( 1- \frac{ k }{ 2 } \right)$$

Whakaoti mō k

$$k=2$$

$$k=-\frac{2}{3}\approx -0.666666667$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakareatia ngā taha e rua o te whārite ki te 2.

$$1\left(1-\frac{k}{2}\right)\left(2-k\right)=2\left(k+2\right)\left(1-\frac{k}{2}\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 1 ki te 1-\frac{k}{2}.

$$\left(1-\frac{k}{2}\right)\left(2-k\right)=2\left(k+2\right)\left(1-\frac{k}{2}\right)$$

Me hoatu te āhuatanga tohatoha mā te whakarea ia tau o 1-\frac{k}{2} ki ia tau o 2-k.

$$2-k+2\left(-\frac{k}{2}\right)-\left(-\frac{k}{2}\right)k=2\left(k+2\right)\left(1-\frac{k}{2}\right)$$

Tuhia te 2\left(-\frac{k}{2}\right) hei hautanga kotahi.

$$2-k+\frac{-2k}{2}-\left(-\frac{k}{2}\right)k=2\left(k+2\right)\left(1-\frac{k}{2}\right)$$

Me whakakore te 2 me te 2.

$$2-k-k-\left(-\frac{k}{2}\right)k=2\left(k+2\right)\left(1-\frac{k}{2}\right)$$

Pahekotia te -k me -k, ka -2k.

$$2-2k-\left(-\frac{k}{2}\right)k=2\left(k+2\right)\left(1-\frac{k}{2}\right)$$

Whakareatia te -1 ki te -1, ka 1.

$$2-2k+\frac{k}{2}k=2\left(k+2\right)\left(1-\frac{k}{2}\right)$$

Tuhia te \frac{k}{2}k hei hautanga kotahi.

$$2-2k+\frac{kk}{2}=2\left(k+2\right)\left(1-\frac{k}{2}\right)$$

Whakareatia te k ki te k, ka k^{2}.

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

Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te k+2.

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

Me hoatu te āhuatanga tohatoha mā te whakarea ia tau o 2k+4 ki ia tau o 1-\frac{k}{2}.

$$2-2k+\frac{k^{2}}{2}=2k+2k\left(-\frac{k}{2}\right)+4+4\left(-\frac{k}{2}\right)$$

Tuhia te 2\left(-\frac{k}{2}\right) hei hautanga kotahi.

$$2-2k+\frac{k^{2}}{2}=2k+\frac{-2k}{2}k+4+4\left(-\frac{k}{2}\right)$$

Me whakakore te 2 me te 2.

$$2-2k+\frac{k^{2}}{2}=2k-kk+4+4\left(-\frac{k}{2}\right)$$

Whakakorea atu te tauwehe pūnoa nui rawa 2 i roto i te 4 me te 2.

$$2-2k+\frac{k^{2}}{2}=2k-kk+4-2k$$

Pahekotia te 2k me -2k, ka 0.

$$2-2k+\frac{k^{2}}{2}=-kk+4$$

Whakareatia te k ki te k, ka k^{2}.

$$2-2k+\frac{k^{2}}{2}=-k^{2}+4$$

Me tāpiri te k^{2} ki ngā taha e rua.

$$2-2k+\frac{k^{2}}{2}+k^{2}=4$$

Pahekotia te \frac{k^{2}}{2} me k^{2}, ka \frac{3}{2}k^{2}.

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

Tangohia te 4 mai i ngā taha e rua.

$$2-2k+\frac{3}{2}k^{2}-4=0$$

Tangohia te 4 i te 2, ka -2.

$$-2-2k+\frac{3}{2}k^{2}=0$$

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.

$$\frac{3}{2}k^{2}-2k-2=0$$

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

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

Pūrua -2.

$$k=\frac{-\left(-2\right)±\sqrt{4-4\times \frac{3}{2}\left(-2\right)}}{2\times \frac{3}{2}}$$

Whakareatia -4 ki te \frac{3}{2}.

$$k=\frac{-\left(-2\right)±\sqrt{4-6\left(-2\right)}}{2\times \frac{3}{2}}$$

Whakareatia -6 ki te -2.

$$k=\frac{-\left(-2\right)±\sqrt{4+12}}{2\times \frac{3}{2}}$$

Tāpiri 4 ki te 12.

$$k=\frac{-\left(-2\right)±\sqrt{16}}{2\times \frac{3}{2}}$$

Tuhia te pūtakerua o te 16.

$$k=\frac{-\left(-2\right)±4}{2\times \frac{3}{2}}$$

Ko te tauaro o -2 ko 2.

$$k=\frac{2±4}{2\times \frac{3}{2}}$$

Whakareatia 2 ki te \frac{3}{2}.

$$k=\frac{2±4}{3}$$

Nā, me whakaoti te whārite k=\frac{2±4}{3} ina he tāpiri te ±. Tāpiri 2 ki te 4.

$$k=\frac{6}{3}$$

Whakawehe 6 ki te 3.

$$k=2$$

Nā, me whakaoti te whārite k=\frac{2±4}{3} ina he tango te ±. Tango 4 mai i 2.

$$k=-\frac{2}{3}$$

Kua oti te whārite te whakatau.

$$k=2 k=-\frac{2}{3}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakareatia ngā taha e rua o te whārite ki te 2.

$$1\left(1-\frac{k}{2}\right)\left(2-k\right)=2\left(k+2\right)\left(1-\frac{k}{2}\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 1 ki te 1-\frac{k}{2}.

$$\left(1-\frac{k}{2}\right)\left(2-k\right)=2\left(k+2\right)\left(1-\frac{k}{2}\right)$$

Me hoatu te āhuatanga tohatoha mā te whakarea ia tau o 1-\frac{k}{2} ki ia tau o 2-k.

$$2-k+2\left(-\frac{k}{2}\right)-\left(-\frac{k}{2}\right)k=2\left(k+2\right)\left(1-\frac{k}{2}\right)$$

Tuhia te 2\left(-\frac{k}{2}\right) hei hautanga kotahi.

$$2-k+\frac{-2k}{2}-\left(-\frac{k}{2}\right)k=2\left(k+2\right)\left(1-\frac{k}{2}\right)$$

Me whakakore te 2 me te 2.

$$2-k-k-\left(-\frac{k}{2}\right)k=2\left(k+2\right)\left(1-\frac{k}{2}\right)$$

Pahekotia te -k me -k, ka -2k.

$$2-2k-\left(-\frac{k}{2}\right)k=2\left(k+2\right)\left(1-\frac{k}{2}\right)$$

Whakareatia te -1 ki te -1, ka 1.

$$2-2k+\frac{k}{2}k=2\left(k+2\right)\left(1-\frac{k}{2}\right)$$

Tuhia te \frac{k}{2}k hei hautanga kotahi.

$$2-2k+\frac{kk}{2}=2\left(k+2\right)\left(1-\frac{k}{2}\right)$$

Whakareatia te k ki te k, ka k^{2}.

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

Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te k+2.

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

Me hoatu te āhuatanga tohatoha mā te whakarea ia tau o 2k+4 ki ia tau o 1-\frac{k}{2}.

$$2-2k+\frac{k^{2}}{2}=2k+2k\left(-\frac{k}{2}\right)+4+4\left(-\frac{k}{2}\right)$$

Tuhia te 2\left(-\frac{k}{2}\right) hei hautanga kotahi.

$$2-2k+\frac{k^{2}}{2}=2k+\frac{-2k}{2}k+4+4\left(-\frac{k}{2}\right)$$

Me whakakore te 2 me te 2.

$$2-2k+\frac{k^{2}}{2}=2k-kk+4+4\left(-\frac{k}{2}\right)$$

Whakakorea atu te tauwehe pūnoa nui rawa 2 i roto i te 4 me te 2.

$$2-2k+\frac{k^{2}}{2}=2k-kk+4-2k$$

Pahekotia te 2k me -2k, ka 0.

$$2-2k+\frac{k^{2}}{2}=-kk+4$$

Whakareatia te k ki te k, ka k^{2}.

$$2-2k+\frac{k^{2}}{2}=-k^{2}+4$$

Me tāpiri te k^{2} ki ngā taha e rua.

$$2-2k+\frac{k^{2}}{2}+k^{2}=4$$

Pahekotia te \frac{k^{2}}{2} me k^{2}, ka \frac{3}{2}k^{2}.

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

Tangohia te 2 mai i ngā taha e rua.

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

Tangohia te 2 i te 4, ka 2.

$$-2k+\frac{3}{2}k^{2}=2$$

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.

$$\frac{3}{2}k^{2}-2k=2$$

Whakawehea ngā taha e rua o te whārite ki te \frac{3}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$\frac{\frac{3}{2}k^{2}-2k}{\frac{3}{2}}=\frac{2}{\frac{3}{2}}$$

Mā te whakawehe ki te \frac{3}{2} ka wetekia te whakareanga ki te \frac{3}{2}.

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

Whakawehe -2 ki te \frac{3}{2} mā te whakarea -2 ki te tau huripoki o \frac{3}{2}.

$$k^{2}-\frac{4}{3}k=\frac{2}{\frac{3}{2}}$$

Whakawehe 2 ki te \frac{3}{2} mā te whakarea 2 ki te tau huripoki o \frac{3}{2}.

$$k^{2}-\frac{4}{3}k=\frac{4}{3}$$

Whakawehea te -\frac{4}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{2}{3}. Nā, tāpiria te pūrua o te -\frac{2}{3} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$k^{2}-\frac{4}{3}k+\left(-\frac{2}{3}\right)^{2}=\frac{4}{3}+\left(-\frac{2}{3}\right)^{2}$$

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

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

Tāpiri \frac{4}{3} ki te \frac{4}{9} 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.

$$k^{2}-\frac{4}{3}k+\frac{4}{9}=\frac{16}{9}$$

Tauwehea k^{2}-\frac{4}{3}k+\frac{4}{9}. 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}.

$$\left(k-\frac{2}{3}\right)^{2}=\frac{16}{9}$$

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

$$\sqrt{\left(k-\frac{2}{3}\right)^{2}}=\sqrt{\frac{16}{9}}$$

Whakarūnātia.

$$k-\frac{2}{3}=\frac{4}{3} k-\frac{2}{3}=-\frac{4}{3}$$

Me tāpiri \frac{2}{3} ki ngā taha e rua o te whārite.

$$k=2 k=-\frac{2}{3}$$