Whakaoti i te x^2+36x-576=0

$$x ^ { 2 } + 36 x - 576 = 0$$

Whakaoti mō x

$$x=-48$$

$$x=12$$

Ngā Upane Whakamahi Tauwehe

Hei whakaoti i te whārite, whakatauwehea te x^{2}+36x-576 mā te whakamahi i te tātai x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=36 ab=-576$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōrunga te a+b, he nui ake te uara pū o te tau tōrunga i tō te tōraro. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -576.

$$-1,576 -2,288 -3,192 -4,144 -6,96 -8,72 -9,64 -12,48 -16,36 -18,32 -24,24$$

Tātaihia te tapeke mō ia takirua.

$$-1+576=575 -2+288=286 -3+192=189 -4+144=140 -6+96=90 -8+72=64 -9+64=55 -12+48=36 -16+36=20 -18+32=14 -24+24=0$$

Ko te otinga te takirua ka hoatu i te tapeke 36.

$$a=-12 b=48$$

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

$$\left(x-12\right)\left(x+48\right)$$

Hei kimi otinga whārite, me whakaoti te x-12=0 me te x+48=0.

$$x=12 x=-48$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx-576. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=36 ab=1\left(-576\right)=-576$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōrunga te a+b, he nui ake te uara pū o te tau tōrunga i tō te tōraro. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -576.

$$-1,576 -2,288 -3,192 -4,144 -6,96 -8,72 -9,64 -12,48 -16,36 -18,32 -24,24$$

Tātaihia te tapeke mō ia takirua.

$$-1+576=575 -2+288=286 -3+192=189 -4+144=140 -6+96=90 -8+72=64 -9+64=55 -12+48=36 -16+36=20 -18+32=14 -24+24=0$$

Ko te otinga te takirua ka hoatu i te tapeke 36.

$$a=-12 b=48$$

Tuhia anō te x^{2}+36x-576 hei \left(x^{2}-12x\right)+\left(48x-576\right).

$$\left(x^{2}-12x\right)+\left(48x-576\right)$$

Tauwehea te x i te tuatahi me te 48 i te rōpū tuarua.

$$x\left(x-12\right)+48\left(x-12\right)$$

Whakatauwehea atu te kīanga pātahi x-12 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-12\right)\left(x+48\right)$$

Hei kimi otinga whārite, me whakaoti te x-12=0 me te x+48=0.

$$x=12 x=-48$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

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.

$$x^{2}+36x-576=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 36 mō b, me -576 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-36±\sqrt{36^{2}-4\left(-576\right)}}{2}$$

Pūrua 36.

$$x=\frac{-36±\sqrt{1296-4\left(-576\right)}}{2}$$

Whakareatia -4 ki te -576.

$$x=\frac{-36±\sqrt{1296+2304}}{2}$$

Tāpiri 1296 ki te 2304.

$$x=\frac{-36±\sqrt{3600}}{2}$$

Tuhia te pūtakerua o te 3600.

$$x=\frac{-36±60}{2}$$

Nā, me whakaoti te whārite x=\frac{-36±60}{2} ina he tāpiri te ±. Tāpiri -36 ki te 60.

$$x=\frac{24}{2}$$

Whakawehe 24 ki te 2.

$$x=12$$

Nā, me whakaoti te whārite x=\frac{-36±60}{2} ina he tango te ±. Tango 60 mai i -36.

$$x=-\frac{96}{2}$$

Whakawehe -96 ki te 2.

$$x=-48$$

Kua oti te whārite te whakatau.

$$x=12 x=-48$$

Ngā Upane mō te Whakaoti i te Pūrua

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.

$$x^{2}+36x-576=0$$

Me tāpiri 576 ki ngā taha e rua o te whārite.

$$x^{2}+36x-576-\left(-576\right)=-\left(-576\right)$$

Mā te tango i te -576 i a ia ake anō ka toe ko te 0.

$$x^{2}+36x=-\left(-576\right)$$

Tango -576 mai i 0.

$$x^{2}+36x=576$$

Whakawehea te 36, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 18. Nā, tāpiria te pūrua o te 18 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+36x+18^{2}=576+18^{2}$$

Pūrua 18.

$$x^{2}+36x+324=576+324$$

Tāpiri 576 ki te 324.

$$x^{2}+36x+324=900$$

Tauwehea x^{2}+36x+324. 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}.

$$\left(x+18\right)^{2}=900$$

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

$$\sqrt{\left(x+18\right)^{2}}=\sqrt{900}$$

Whakarūnātia.

$$x+18=30 x+18=-30$$

Me tango 18 mai i ngā taha e rua o te whārite.

$$x=12 x=-48$$