Whakaoti i te x^2+14x-2352=0

$${ x }^{ 2 } +14x-2352=0$$

Whakaoti mō x

$$x=-56$$

$$x=42$$

Ngā Upane Whakamahi Tauwehe

Hei whakaoti i te whārite, whakatauwehea te x^{2}+14x-2352 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.

$$a+b=14 ab=-2352$$

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 -2352.

$$-1,2352 -2,1176 -3,784 -4,588 -6,392 -7,336 -8,294 -12,196 -14,168 -16,147 -21,112 -24,98 -28,84 -42,56 -48,49$$

Tātaihia te tapeke mō ia takirua.

$$-1+2352=2351 -2+1176=1174 -3+784=781 -4+588=584 -6+392=386 -7+336=329 -8+294=286 -12+196=184 -14+168=154 -16+147=131 -21+112=91 -24+98=74 -28+84=56 -42+56=14 -48+49=1$$

Ko te otinga te takirua ka hoatu i te tapeke 14.

$$a=-42 b=56$$

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

$$\left(x-42\right)\left(x+56\right)$$

Hei kimi otinga whārite, me whakaoti te x-42=0 me te x+56=0.

$$x=42 x=-56$$

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-2352. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=14 ab=1\left(-2352\right)=-2352$$

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 -2352.

$$-1,2352 -2,1176 -3,784 -4,588 -6,392 -7,336 -8,294 -12,196 -14,168 -16,147 -21,112 -24,98 -28,84 -42,56 -48,49$$

Tātaihia te tapeke mō ia takirua.

$$-1+2352=2351 -2+1176=1174 -3+784=781 -4+588=584 -6+392=386 -7+336=329 -8+294=286 -12+196=184 -14+168=154 -16+147=131 -21+112=91 -24+98=74 -28+84=56 -42+56=14 -48+49=1$$

Ko te otinga te takirua ka hoatu i te tapeke 14.

$$a=-42 b=56$$

Tuhia anō te x^{2}+14x-2352 hei \left(x^{2}-42x\right)+\left(56x-2352\right).

$$\left(x^{2}-42x\right)+\left(56x-2352\right)$$

Tauwehea te x i te tuatahi me te 56 i te rōpū tuarua.

$$x\left(x-42\right)+56\left(x-42\right)$$

Whakatauwehea atu te kīanga pātahi x-42 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-42\right)\left(x+56\right)$$

Hei kimi otinga whārite, me whakaoti te x-42=0 me te x+56=0.

$$x=42 x=-56$$

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.

$$x^{2}+14x-2352=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 14 mō b, me -2352 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-14±\sqrt{14^{2}-4\left(-2352\right)}}{2}$$

Pūrua 14.

$$x=\frac{-14±\sqrt{196-4\left(-2352\right)}}{2}$$

Whakareatia -4 ki te -2352.

$$x=\frac{-14±\sqrt{196+9408}}{2}$$

Tāpiri 196 ki te 9408.

$$x=\frac{-14±\sqrt{9604}}{2}$$

Tuhia te pūtakerua o te 9604.

$$x=\frac{-14±98}{2}$$

Nā, me whakaoti te whārite x=\frac{-14±98}{2} ina he tāpiri te ±. Tāpiri -14 ki te 98.

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

Whakawehe 84 ki te 2.

$$x=42$$

Nā, me whakaoti te whārite x=\frac{-14±98}{2} ina he tango te ±. Tango 98 mai i -14.

$$x=-\frac{112}{2}$$

Whakawehe -112 ki te 2.

$$x=-56$$

Kua oti te whārite te whakatau.

$$x=42 x=-56$$

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.

$$x^{2}+14x-2352=0$$

Me tāpiri 2352 ki ngā taha e rua o te whārite.

$$x^{2}+14x-2352-\left(-2352\right)=-\left(-2352\right)$$

Mā te tango i te -2352 i a ia ake anō ka toe ko te 0.

$$x^{2}+14x=-\left(-2352\right)$$

Tango -2352 mai i 0.

$$x^{2}+14x=2352$$

Whakawehea te 14, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 7. Nā, tāpiria te pūrua o te 7 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+14x+7^{2}=2352+7^{2}$$

Pūrua 7.

$$x^{2}+14x+49=2352+49$$

Tāpiri 2352 ki te 49.

$$x^{2}+14x+49=2401$$

Tauwehea x^{2}+14x+49. 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}.

$$\left(x+7\right)^{2}=2401$$

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

$$\sqrt{\left(x+7\right)^{2}}=\sqrt{2401}$$

Whakarūnātia.

$$x+7=49 x+7=-49$$

Me tango 7 mai i ngā taha e rua o te whārite.

$$x=42 x=-56$$