Whakaoti i te 16k^2+24k^4/(2k^2+1)^2=20/9

$$\frac { 16 k ^ { 2 } + 24 k ^ { 4 } } { ( 2 k ^ { 2 } + 1 ) ^ { 2 } } = \frac { 20 } { 9 }$$

Whakaoti mō k

$$k=-\frac{\sqrt{102\sqrt{26}-272}}{34}\approx -0.463270298$$

$$k=\frac{\sqrt{102\sqrt{26}-272}}{34}\approx 0.463270298$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Me whakarea ngā taha e rua o te whārite ki te 9\left(2k^{2}+1\right)^{2}, arā, te tauraro pātahi he tino iti rawa te kitea o \left(2k^{2}+1\right)^{2},9.

$$9\left(16k^{2}+24k^{4}\right)=20\left(2k^{2}+1\right)^{2}$$

Whakamahia te āhuatanga tohatoha hei whakarea te 9 ki te 16k^{2}+24k^{4}.

$$144k^{2}+216k^{4}=20\left(2k^{2}+1\right)^{2}$$

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

$$144k^{2}+216k^{4}=20\left(4\left(k^{2}\right)^{2}+4k^{2}+1\right)$$

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

$$144k^{2}+216k^{4}=20\left(4k^{4}+4k^{2}+1\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 20 ki te 4k^{4}+4k^{2}+1.

$$144k^{2}+216k^{4}=80k^{4}+80k^{2}+20$$

Tangohia te 80k^{4} mai i ngā taha e rua.

$$144k^{2}+216k^{4}-80k^{4}=80k^{2}+20$$

Pahekotia te 216k^{4} me -80k^{4}, ka 136k^{4}.

$$144k^{2}+136k^{4}=80k^{2}+20$$

Tangohia te 80k^{2} mai i ngā taha e rua.

$$144k^{2}+136k^{4}-80k^{2}=20$$

Pahekotia te 144k^{2} me -80k^{2}, ka 64k^{2}.

$$64k^{2}+136k^{4}=20$$

Tangohia te 20 mai i ngā taha e rua.

$$64k^{2}+136k^{4}-20=0$$

Whakakapia te t mō te k^{2}.

$$136t^{2}+64t-20=0$$

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 136 mō te a, te 64 mō te b, me te -20 mō te c i te ture pūrua.

$$t=\frac{-64±\sqrt{64^{2}-4\times 136\left(-20\right)}}{2\times 136}$$

Mahia ngā tātaitai.

$$t=\frac{-64±24\sqrt{26}}{272}$$

Whakaotia te whārite t=\frac{-64±24\sqrt{26}}{272} ina he tōrunga te ±, ina he tōraro te ±.

$$t=\frac{3\sqrt{26}}{34}-\frac{4}{17} t=-\frac{3\sqrt{26}}{34}-\frac{4}{17}$$

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

$$k=\frac{\sqrt{\frac{6\sqrt{26}-16}{17}}}{2} k=-\frac{\sqrt{\frac{6\sqrt{26}-16}{17}}}{2}$$