Whakaoti i te k^2-4(8k+36)=0

$$k ^ { 2 } - 4 ( 8 k + 36 ) = 0$$

Whakaoti mō k

$$k=-4$$

$$k=36$$

Ngā Upane Whakamahi Tauwehe

Whakamahia te āhuatanga tohatoha hei whakarea te -4 ki te 8k+36.

$$k^{2}-32k-144=0$$

Hei whakaoti i te whārite, whakatauwehea te k^{2}-32k-144 mā te whakamahi i te tātai k^{2}+\left(a+b\right)k+ab=\left(k+a\right)\left(k+b\right). Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-32 ab=-144$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōraro te a+b, he nui ake te uara pū o te tau tōraro i tō te tōrunga. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -144.

$$1,-144 2,-72 3,-48 4,-36 6,-24 8,-18 9,-16 12,-12$$

Tātaihia te tapeke mō ia takirua.

$$1-144=-143 2-72=-70 3-48=-45 4-36=-32 6-24=-18 8-18=-10 9-16=-7 12-12=0$$

Ko te otinga te takirua ka hoatu i te tapeke -32.

$$a=-36 b=4$$

Me tuhi anō te kīanga whakatauwehe \left(k+a\right)\left(k+b\right) mā ngā uara i tātaihia.

$$\left(k-36\right)\left(k+4\right)$$

Hei kimi otinga whārite, me whakaoti te k-36=0 me te k+4=0.

$$k=36 k=-4$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakamahia te āhuatanga tohatoha hei whakarea te -4 ki te 8k+36.

$$k^{2}-32k-144=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei k^{2}+ak+bk-144. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-32 ab=1\left(-144\right)=-144$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōraro te a+b, he nui ake te uara pū o te tau tōraro i tō te tōrunga. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -144.

$$1,-144 2,-72 3,-48 4,-36 6,-24 8,-18 9,-16 12,-12$$

Tātaihia te tapeke mō ia takirua.

$$1-144=-143 2-72=-70 3-48=-45 4-36=-32 6-24=-18 8-18=-10 9-16=-7 12-12=0$$

Ko te otinga te takirua ka hoatu i te tapeke -32.

$$a=-36 b=4$$

Tuhia anō te k^{2}-32k-144 hei \left(k^{2}-36k\right)+\left(4k-144\right).

$$\left(k^{2}-36k\right)+\left(4k-144\right)$$

Tauwehea te k i te tuatahi me te 4 i te rōpū tuarua.

$$k\left(k-36\right)+4\left(k-36\right)$$

Whakatauwehea atu te kīanga pātahi k-36 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(k-36\right)\left(k+4\right)$$

Hei kimi otinga whārite, me whakaoti te k-36=0 me te k+4=0.

$$k=36 k=-4$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tohatoha hei whakarea te -4 ki te 8k+36.

$$k^{2}-32k-144=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -32 mō b, me -144 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$k=\frac{-\left(-32\right)±\sqrt{\left(-32\right)^{2}-4\left(-144\right)}}{2}$$

Pūrua -32.

$$k=\frac{-\left(-32\right)±\sqrt{1024-4\left(-144\right)}}{2}$$

Whakareatia -4 ki te -144.

$$k=\frac{-\left(-32\right)±\sqrt{1024+576}}{2}$$

Tāpiri 1024 ki te 576.

$$k=\frac{-\left(-32\right)±\sqrt{1600}}{2}$$

Tuhia te pūtakerua o te 1600.

$$k=\frac{-\left(-32\right)±40}{2}$$

Ko te tauaro o -32 ko 32.

$$k=\frac{32±40}{2}$$

Nā, me whakaoti te whārite k=\frac{32±40}{2} ina he tāpiri te ±. Tāpiri 32 ki te 40.

$$k=\frac{72}{2}$$

Whakawehe 72 ki te 2.

$$k=36$$

Nā, me whakaoti te whārite k=\frac{32±40}{2} ina he tango te ±. Tango 40 mai i 32.

$$k=-\frac{8}{2}$$

Whakawehe -8 ki te 2.

$$k=-4$$

Kua oti te whārite te whakatau.

$$k=36 k=-4$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tohatoha hei whakarea te -4 ki te 8k+36.

$$k^{2}-32k-144=0$$

Me tāpiri te 144 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$k^{2}-32k=144$$

Whakawehea te -32, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -16. Nā, tāpiria te pūrua o te -16 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$k^{2}-32k+\left(-16\right)^{2}=144+\left(-16\right)^{2}$$

Pūrua -16.

$$k^{2}-32k+256=144+256$$

Tāpiri 144 ki te 256.

$$k^{2}-32k+256=400$$

Tauwehea k^{2}-32k+256. 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}.

$$\left(k-16\right)^{2}=400$$

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

$$\sqrt{\left(k-16\right)^{2}}=\sqrt{400}$$

Whakarūnātia.

$$k-16=20 k-16=-20$$

Me tāpiri 16 ki ngā taha e rua o te whārite.

$$k=36 k=-4$$