Whakaoti i te (-12-k)^2-4(4)(4)=0

$$( - 12 - k ) ^ { 2 } - 4 ( 4 ) ( 4 ) = 0$$

Whakaoti mō k

$$k=-20$$

$$k=-4$$

Ngā Upane Whakamahi Tauwehe

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(-12-k\right)^{2}.

$$144+24k+k^{2}-4\times 4\times 4=0$$

Whakareatia te 4 ki te 4, ka 16.

$$144+24k+k^{2}-16\times 4=0$$

Whakareatia te 16 ki te 4, ka 64.

$$144+24k+k^{2}-64=0$$

Tangohia te 64 i te 144, ka 80.

$$80+24k+k^{2}=0$$

Hurinahatia te pūrau ki te āhua tānga ngahuru. Whakaraupapahia ngā kīanga tau mai i te pū teitei rawa ki te mea iti rawa.

$$k^{2}+24k+80=0$$

Hei whakaoti i te whārite, whakatauwehea te k^{2}+24k+80 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.

$$a+b=24 ab=80$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōrunga te a+b, he tōrunga hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 80.

$$1,80 2,40 4,20 5,16 8,10$$

Tātaihia te tapeke mō ia takirua.

$$1+80=81 2+40=42 4+20=24 5+16=21 8+10=18$$

Ko te otinga te takirua ka hoatu i te tapeke 24.

$$a=4 b=20$$

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

$$\left(k+4\right)\left(k+20\right)$$

Hei kimi otinga whārite, me whakaoti te k+4=0 me te k+20=0.

$$k=-4 k=-20$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(-12-k\right)^{2}.

$$144+24k+k^{2}-4\times 4\times 4=0$$

Whakareatia te 4 ki te 4, ka 16.

$$144+24k+k^{2}-16\times 4=0$$

Whakareatia te 16 ki te 4, ka 64.

$$144+24k+k^{2}-64=0$$

Tangohia te 64 i te 144, ka 80.

$$80+24k+k^{2}=0$$

Hurinahatia te pūrau ki te āhua tānga ngahuru. Whakaraupapahia ngā kīanga tau mai i te pū teitei rawa ki te mea iti rawa.

$$k^{2}+24k+80=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei k^{2}+ak+bk+80. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=24 ab=1\times 80=80$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōrunga te a+b, he tōrunga hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 80.

$$1,80 2,40 4,20 5,16 8,10$$

Tātaihia te tapeke mō ia takirua.

$$1+80=81 2+40=42 4+20=24 5+16=21 8+10=18$$

Ko te otinga te takirua ka hoatu i te tapeke 24.

$$a=4 b=20$$

Tuhia anō te k^{2}+24k+80 hei \left(k^{2}+4k\right)+\left(20k+80\right).

$$\left(k^{2}+4k\right)+\left(20k+80\right)$$

Tauwehea te k i te tuatahi me te 20 i te rōpū tuarua.

$$k\left(k+4\right)+20\left(k+4\right)$$

Whakatauwehea atu te kīanga pātahi k+4 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(k+4\right)\left(k+20\right)$$

Hei kimi otinga whārite, me whakaoti te k+4=0 me te k+20=0.

$$k=-4 k=-20$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(-12-k\right)^{2}.

$$144+24k+k^{2}-4\times 4\times 4=0$$

Whakareatia te 4 ki te 4, ka 16.

$$144+24k+k^{2}-16\times 4=0$$

Whakareatia te 16 ki te 4, ka 64.

$$144+24k+k^{2}-64=0$$

Tangohia te 64 i te 144, ka 80.

$$80+24k+k^{2}=0$$

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.

$$k^{2}+24k+80=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 24 mō b, me 80 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$k=\frac{-24±\sqrt{24^{2}-4\times 80}}{2}$$

Pūrua 24.

$$k=\frac{-24±\sqrt{576-4\times 80}}{2}$$

Whakareatia -4 ki te 80.

$$k=\frac{-24±\sqrt{576-320}}{2}$$

Tāpiri 576 ki te -320.

$$k=\frac{-24±\sqrt{256}}{2}$$

Tuhia te pūtakerua o te 256.

$$k=\frac{-24±16}{2}$$

Nā, me whakaoti te whārite k=\frac{-24±16}{2} ina he tāpiri te ±. Tāpiri -24 ki te 16.

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

Whakawehe -8 ki te 2.

$$k=-4$$

Nā, me whakaoti te whārite k=\frac{-24±16}{2} ina he tango te ±. Tango 16 mai i -24.

$$k=-\frac{40}{2}$$

Whakawehe -40 ki te 2.

$$k=-20$$

Kua oti te whārite te whakatau.

$$k=-4 k=-20$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(-12-k\right)^{2}.

$$144+24k+k^{2}-4\times 4\times 4=0$$

Whakareatia te 4 ki te 4, ka 16.

$$144+24k+k^{2}-16\times 4=0$$

Whakareatia te 16 ki te 4, ka 64.

$$144+24k+k^{2}-64=0$$

Tangohia te 64 i te 144, ka 80.

$$80+24k+k^{2}=0$$

Tangohia te 80 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$24k+k^{2}=-80$$

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.

$$k^{2}+24k=-80$$

Whakawehea te 24, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 12. Nā, tāpiria te pūrua o te 12 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$k^{2}+24k+12^{2}=-80+12^{2}$$

Pūrua 12.

$$k^{2}+24k+144=-80+144$$

Tāpiri -80 ki te 144.

$$k^{2}+24k+144=64$$

Tauwehea k^{2}+24k+144. 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}.

$$\left(k+12\right)^{2}=64$$

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

$$\sqrt{\left(k+12\right)^{2}}=\sqrt{64}$$

Whakarūnātia.

$$k+12=8 k+12=-8$$

Me tango 12 mai i ngā taha e rua o te whārite.

$$k=-4 k=-20$$