Whakaoti i te frac{6(k^{2}+1)^{2}-(3k^2-1)^2}{(3k^2+1)^2}=5/4

$$\frac { 6 ( k ^ { 2 } + 1 ) ^ { 2 } - ( 3 k ^ { 2 } - 1 ) ^ { 2 } } { ( 3 k ^ { 2 } + 1 ) ^ { 2 } } = \frac { 5 } { 4 }$$

Whakaoti mō k

$$k=-1$$

$$k=1$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Me whakarea ngā taha e rua o te whārite ki te 4\left(3k^{2}+1\right)^{2}, arā, te tauraro pātahi he tino iti rawa te kitea o \left(3k^{2}+1\right)^{2},4.

$$4\left(6\left(k^{2}+1\right)^{2}-\left(3k^{2}-1\right)^{2}\right)=5\left(3k^{2}+1\right)^{2}$$

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

$$4\left(6\left(\left(k^{2}\right)^{2}+2k^{2}+1\right)-\left(3k^{2}-1\right)^{2}\right)=5\left(3k^{2}+1\right)^{2}$$

Hei hiki pū ki tētahi pū anō, me whakarea ngā taupū. Me whakarea te 2 me te 2 kia riro ai te 4.

$$4\left(6\left(k^{4}+2k^{2}+1\right)-\left(3k^{2}-1\right)^{2}\right)=5\left(3k^{2}+1\right)^{2}$$

Whakamahia te āhuatanga tohatoha hei whakarea te 6 ki te k^{4}+2k^{2}+1.

$$4\left(6k^{4}+12k^{2}+6-\left(3k^{2}-1\right)^{2}\right)=5\left(3k^{2}+1\right)^{2}$$

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(3k^{2}-1\right)^{2}.

$$4\left(6k^{4}+12k^{2}+6-\left(9\left(k^{2}\right)^{2}-6k^{2}+1\right)\right)=5\left(3k^{2}+1\right)^{2}$$

Hei hiki pū ki tētahi pū anō, me whakarea ngā taupū. Me whakarea te 2 me te 2 kia riro ai te 4.

$$4\left(6k^{4}+12k^{2}+6-\left(9k^{4}-6k^{2}+1\right)\right)=5\left(3k^{2}+1\right)^{2}$$

Hei kimi i te tauaro o 9k^{4}-6k^{2}+1, kimihia te tauaro o ia taurangi.

$$4\left(6k^{4}+12k^{2}+6-9k^{4}+6k^{2}-1\right)=5\left(3k^{2}+1\right)^{2}$$

Pahekotia te 6k^{4} me -9k^{4}, ka -3k^{4}.

$$4\left(-3k^{4}+12k^{2}+6+6k^{2}-1\right)=5\left(3k^{2}+1\right)^{2}$$

Pahekotia te 12k^{2} me 6k^{2}, ka 18k^{2}.

$$4\left(-3k^{4}+18k^{2}+6-1\right)=5\left(3k^{2}+1\right)^{2}$$

Tangohia te 1 i te 6, ka 5.

$$4\left(-3k^{4}+18k^{2}+5\right)=5\left(3k^{2}+1\right)^{2}$$

Whakamahia te āhuatanga tohatoha hei whakarea te 4 ki te -3k^{4}+18k^{2}+5.

$$-12k^{4}+72k^{2}+20=5\left(3k^{2}+1\right)^{2}$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(3k^{2}+1\right)^{2}.

$$-12k^{4}+72k^{2}+20=5\left(9\left(k^{2}\right)^{2}+6k^{2}+1\right)$$

Hei hiki pū ki tētahi pū anō, me whakarea ngā taupū. Me whakarea te 2 me te 2 kia riro ai te 4.

$$-12k^{4}+72k^{2}+20=5\left(9k^{4}+6k^{2}+1\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 5 ki te 9k^{4}+6k^{2}+1.

$$-12k^{4}+72k^{2}+20=45k^{4}+30k^{2}+5$$

Tangohia te 45k^{4} mai i ngā taha e rua.

$$-12k^{4}+72k^{2}+20-45k^{4}=30k^{2}+5$$

Pahekotia te -12k^{4} me -45k^{4}, ka -57k^{4}.

$$-57k^{4}+72k^{2}+20=30k^{2}+5$$

Tangohia te 30k^{2} mai i ngā taha e rua.

$$-57k^{4}+72k^{2}+20-30k^{2}=5$$

Pahekotia te 72k^{2} me -30k^{2}, ka 42k^{2}.

$$-57k^{4}+42k^{2}+20=5$$

Tangohia te 5 mai i ngā taha e rua.

$$-57k^{4}+42k^{2}+20-5=0$$

Tangohia te 5 i te 20, ka 15.

$$-57k^{4}+42k^{2}+15=0$$

Whakakapia te t mō te k^{2}.

$$-57t^{2}+42t+15=0$$

Ka taea ngā whārite katoa o te momo ax^{2}+bx+c=0 te whakaoti mā te ture pūrua: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Whakakapia te -57 mō te a, te 42 mō te b, me te 15 mō te c i te ture pūrua.

$$t=\frac{-42±\sqrt{42^{2}-4\left(-57\right)\times 15}}{-57\times 2}$$

Mahia ngā tātaitai.

$$t=\frac{-42±72}{-114}$$

Whakaotia te whārite t=\frac{-42±72}{-114} ina he tōrunga te ±, ina he tōraro te ±.

$$t=-\frac{5}{19} t=1$$

I te mea ko k=t^{2}, ka riro ngā otinga mā te arotake i te k=±\sqrt{t} mō t tōrunga.

$$k=1 k=-1$$