Whakaoti i te x^2+54x=-504

$$x ^ { 2 } + 54 x = - 504$$

Whakaoti mō x

$$x=-42$$

$$x=-12$$

Ngā Upane Whakamahi Tauwehe

Me tāpiri te 504 ki ngā taha e rua.

$$x^{2}+54x+504=0$$

Hei whakaoti i te whārite, whakatauwehea te x^{2}+54x+504 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.

$$a+b=54 ab=504$$

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 504.

$$1,504 2,252 3,168 4,126 6,84 7,72 8,63 9,56 12,42 14,36 18,28 21,24$$

Tātaihia te tapeke mō ia takirua.

$$1+504=505 2+252=254 3+168=171 4+126=130 6+84=90 7+72=79 8+63=71 9+56=65 12+42=54 14+36=50 18+28=46 21+24=45$$

Ko te otinga te takirua ka hoatu i te tapeke 54.

$$a=12 b=42$$

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

$$\left(x+12\right)\left(x+42\right)$$

Hei kimi otinga whārite, me whakaoti te x+12=0 me te x+42=0.

$$x=-12 x=-42$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Me tāpiri te 504 ki ngā taha e rua.

$$x^{2}+54x+504=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx+504. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=54 ab=1\times 504=504$$

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 504.

$$1,504 2,252 3,168 4,126 6,84 7,72 8,63 9,56 12,42 14,36 18,28 21,24$$

Tātaihia te tapeke mō ia takirua.

$$1+504=505 2+252=254 3+168=171 4+126=130 6+84=90 7+72=79 8+63=71 9+56=65 12+42=54 14+36=50 18+28=46 21+24=45$$

Ko te otinga te takirua ka hoatu i te tapeke 54.

$$a=12 b=42$$

Tuhia anō te x^{2}+54x+504 hei \left(x^{2}+12x\right)+\left(42x+504\right).

$$\left(x^{2}+12x\right)+\left(42x+504\right)$$

Tauwehea te x i te tuatahi me te 42 i te rōpū tuarua.

$$x\left(x+12\right)+42\left(x+12\right)$$

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

$$\left(x+12\right)\left(x+42\right)$$

Hei kimi otinga whārite, me whakaoti te x+12=0 me te x+42=0.

$$x=-12 x=-42$$

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.

$$x^{2}+54x=-504$$

Me tāpiri 504 ki ngā taha e rua o te whārite.

$$x^{2}+54x-\left(-504\right)=-504-\left(-504\right)$$

Mā te tango i te -504 i a ia ake anō ka toe ko te 0.

$$x^{2}+54x-\left(-504\right)=0$$

Tango -504 mai i 0.

$$x^{2}+54x+504=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 54 mō b, me 504 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-54±\sqrt{54^{2}-4\times 504}}{2}$$

Pūrua 54.

$$x=\frac{-54±\sqrt{2916-4\times 504}}{2}$$

Whakareatia -4 ki te 504.

$$x=\frac{-54±\sqrt{2916-2016}}{2}$$

Tāpiri 2916 ki te -2016.

$$x=\frac{-54±\sqrt{900}}{2}$$

Tuhia te pūtakerua o te 900.

$$x=\frac{-54±30}{2}$$

Nā, me whakaoti te whārite x=\frac{-54±30}{2} ina he tāpiri te ±. Tāpiri -54 ki te 30.

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

Whakawehe -24 ki te 2.

$$x=-12$$

Nā, me whakaoti te whārite x=\frac{-54±30}{2} ina he tango te ±. Tango 30 mai i -54.

$$x=-\frac{84}{2}$$

Whakawehe -84 ki te 2.

$$x=-42$$

Kua oti te whārite te whakatau.

$$x=-12 x=-42$$

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.

$$x^{2}+54x=-504$$

Whakawehea te 54, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 27. Nā, tāpiria te pūrua o te 27 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+54x+27^{2}=-504+27^{2}$$

Pūrua 27.

$$x^{2}+54x+729=-504+729$$

Tāpiri -504 ki te 729.

$$x^{2}+54x+729=225$$

Tauwehea x^{2}+54x+729. 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}.

$$\left(x+27\right)^{2}=225$$

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

$$\sqrt{\left(x+27\right)^{2}}=\sqrt{225}$$

Whakarūnātia.

$$x+27=15 x+27=-15$$

Me tango 27 mai i ngā taha e rua o te whārite.

$$x=-12 x=-42$$