Whakaoti i te 3(-16k/4k^2+1)^2=32/4k^2+1

$$3 ( \frac { - 16 k } { 4 k ^ { 2 } + 1 } ) ^ { 2 } = \frac { 32 } { 4 k ^ { 2 } + 1 }$$

Whakaoti mō k

$$k=\frac{\sqrt{5}}{10}\approx 0.223606798$$

$$k=-\frac{\sqrt{5}}{10}\approx -0.223606798$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakareatia ngā taha e rua o te whārite ki te 4k^{2}+1.

$$3\times \left(\frac{-16k}{4k^{2}+1}\right)^{2}\left(4k^{2}+1\right)=32$$

Kia whakarewa i te \frac{-16k}{4k^{2}+1} ki tētahi taupū, me whakarewa tahi te taurunga me te tauraro ki te taupū kātahi ka whakawehe.

$$3\times \frac{\left(-16k\right)^{2}}{\left(4k^{2}+1\right)^{2}}\left(4k^{2}+1\right)=32$$

Tuhia te 3\times \frac{\left(-16k\right)^{2}}{\left(4k^{2}+1\right)^{2}} hei hautanga kotahi.

$$\frac{3\left(-16k\right)^{2}}{\left(4k^{2}+1\right)^{2}}\left(4k^{2}+1\right)=32$$

Tuhia te \frac{3\left(-16k\right)^{2}}{\left(4k^{2}+1\right)^{2}}\left(4k^{2}+1\right) hei hautanga kotahi.

$$\frac{3\left(-16k\right)^{2}\left(4k^{2}+1\right)}{\left(4k^{2}+1\right)^{2}}=32$$

Whakarohaina te \left(-16k\right)^{2}.

$$\frac{3\left(-16\right)^{2}k^{2}\left(4k^{2}+1\right)}{\left(4k^{2}+1\right)^{2}}=32$$

Tātaihia te -16 mā te pū o 2, kia riro ko 256.

$$\frac{3\times 256k^{2}\left(4k^{2}+1\right)}{\left(4k^{2}+1\right)^{2}}=32$$

Whakareatia te 3 ki te 256, ka 768.

$$\frac{768k^{2}\left(4k^{2}+1\right)}{\left(4k^{2}+1\right)^{2}}=32$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(4k^{2}+1\right)^{2}.

$$\frac{768k^{2}\left(4k^{2}+1\right)}{16\left(k^{2}\right)^{2}+8k^{2}+1}=32$$

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

$$\frac{768k^{2}\left(4k^{2}+1\right)}{16k^{4}+8k^{2}+1}=32$$

Tangohia te 32 mai i ngā taha e rua.

$$\frac{768k^{2}\left(4k^{2}+1\right)}{16k^{4}+8k^{2}+1}-32=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te 768k^{2} ki te 4k^{2}+1.

$$\frac{3072k^{4}+768k^{2}}{16k^{4}+8k^{2}+1}-32=0$$

Tauwehea te 16k^{4}+8k^{2}+1.

$$\frac{3072k^{4}+768k^{2}}{\left(4k^{2}+1\right)^{2}}-32=0$$

Hei tāpiri, hei tango kīanga rānei, me whakaroha ērā kia rite ā rātou tauraro. Whakareatia 32 ki te \frac{\left(4k^{2}+1\right)^{2}}{\left(4k^{2}+1\right)^{2}}.

$$\frac{3072k^{4}+768k^{2}}{\left(4k^{2}+1\right)^{2}}-\frac{32\left(4k^{2}+1\right)^{2}}{\left(4k^{2}+1\right)^{2}}=0$$

Tā te mea he rite te tauraro o \frac{3072k^{4}+768k^{2}}{\left(4k^{2}+1\right)^{2}} me \frac{32\left(4k^{2}+1\right)^{2}}{\left(4k^{2}+1\right)^{2}}, me tango rāua mā te tango i ō raua taurunga.

$$\frac{3072k^{4}+768k^{2}-32\left(4k^{2}+1\right)^{2}}{\left(4k^{2}+1\right)^{2}}=0$$

Mahia ngā whakarea i roto o 3072k^{4}+768k^{2}-32\left(4k^{2}+1\right)^{2}.

$$\frac{3072k^{4}+768k^{2}-512k^{4}-256k^{2}-32}{\left(4k^{2}+1\right)^{2}}=0$$

Whakakotahitia ngā kupu rite i 3072k^{4}+768k^{2}-512k^{4}-256k^{2}-32.

$$\frac{2560k^{4}+512k^{2}-32}{\left(4k^{2}+1\right)^{2}}=0$$

Whakareatia ngā taha e rua o te whārite ki te \left(4k^{2}+1\right)^{2}.

$$2560k^{4}+512k^{2}-32=0$$

Whakakapia te t mō te k^{2}.

$$2560t^{2}+512t-32=0$$

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 2560 mō te a, te 512 mō te b, me te -32 mō te c i te ture pūrua.

$$t=\frac{-512±\sqrt{512^{2}-4\times 2560\left(-32\right)}}{2\times 2560}$$

Mahia ngā tātaitai.

$$t=\frac{-512±768}{5120}$$

Whakaotia te whārite t=\frac{-512±768}{5120} ina he tōrunga te ±, ina he tōraro te ±.

$$t=\frac{1}{20} t=-\frac{1}{4}$$

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

$$k=\frac{\sqrt{5}}{10} k=-\frac{\sqrt{5}}{10}$$