Whakaoti i te x^2+x-342=0

$$x ^ { 2 } + x - 342 = 0$$

Whakaoti mō x

$$x=-19$$

$$x=18$$

Ngā Upane Whakamahi Tauwehe

Hei whakaoti i te whārite, whakatauwehea te x^{2}+x-342 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.

$$a+b=1 ab=-342$$

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 -342.

$$-1,342 -2,171 -3,114 -6,57 -9,38 -18,19$$

Tātaihia te tapeke mō ia takirua.

$$-1+342=341 -2+171=169 -3+114=111 -6+57=51 -9+38=29 -18+19=1$$

Ko te otinga te takirua ka hoatu i te tapeke 1.

$$a=-18 b=19$$

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

$$\left(x-18\right)\left(x+19\right)$$

Hei kimi otinga whārite, me whakaoti te x-18=0 me te x+19=0.

$$x=18 x=-19$$

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-342. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=1 ab=1\left(-342\right)=-342$$

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 -342.

$$-1,342 -2,171 -3,114 -6,57 -9,38 -18,19$$

Tātaihia te tapeke mō ia takirua.

$$-1+342=341 -2+171=169 -3+114=111 -6+57=51 -9+38=29 -18+19=1$$

Ko te otinga te takirua ka hoatu i te tapeke 1.

$$a=-18 b=19$$

Tuhia anō te x^{2}+x-342 hei \left(x^{2}-18x\right)+\left(19x-342\right).

$$\left(x^{2}-18x\right)+\left(19x-342\right)$$

Tauwehea te x i te tuatahi me te 19 i te rōpū tuarua.

$$x\left(x-18\right)+19\left(x-18\right)$$

Whakatauwehea atu te kīanga pātahi x-18 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-18\right)\left(x+19\right)$$

Hei kimi otinga whārite, me whakaoti te x-18=0 me te x+19=0.

$$x=18 x=-19$$

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.

$$x^{2}+x-342=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 1 mō b, me -342 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1±\sqrt{1^{2}-4\left(-342\right)}}{2}$$

Pūrua 1.

$$x=\frac{-1±\sqrt{1-4\left(-342\right)}}{2}$$

Whakareatia -4 ki te -342.

$$x=\frac{-1±\sqrt{1+1368}}{2}$$

Tāpiri 1 ki te 1368.

$$x=\frac{-1±\sqrt{1369}}{2}$$

Tuhia te pūtakerua o te 1369.

$$x=\frac{-1±37}{2}$$

Nā, me whakaoti te whārite x=\frac{-1±37}{2} ina he tāpiri te ±. Tāpiri -1 ki te 37.

$$x=\frac{36}{2}$$

Whakawehe 36 ki te 2.

$$x=18$$

Nā, me whakaoti te whārite x=\frac{-1±37}{2} ina he tango te ±. Tango 37 mai i -1.

$$x=-\frac{38}{2}$$

Whakawehe -38 ki te 2.

$$x=-19$$

Kua oti te whārite te whakatau.

$$x=18 x=-19$$

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.

$$x^{2}+x-342=0$$

Me tāpiri 342 ki ngā taha e rua o te whārite.

$$x^{2}+x-342-\left(-342\right)=-\left(-342\right)$$

Mā te tango i te -342 i a ia ake anō ka toe ko te 0.

$$x^{2}+x=-\left(-342\right)$$

Tango -342 mai i 0.

$$x^{2}+x=342$$

Whakawehea te 1, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{1}{2}. Nā, tāpiria te pūrua o te \frac{1}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+x+\left(\frac{1}{2}\right)^{2}=342+\left(\frac{1}{2}\right)^{2}$$

Pūruatia \frac{1}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+x+\frac{1}{4}=342+\frac{1}{4}$$

Tāpiri 342 ki te \frac{1}{4}.

$$x^{2}+x+\frac{1}{4}=\frac{1369}{4}$$

Tauwehea x^{2}+x+\frac{1}{4}. 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}.

$$\left(x+\frac{1}{2}\right)^{2}=\frac{1369}{4}$$

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

$$\sqrt{\left(x+\frac{1}{2}\right)^{2}}=\sqrt{\frac{1369}{4}}$$

Whakarūnātia.

$$x+\frac{1}{2}=\frac{37}{2} x+\frac{1}{2}=-\frac{37}{2}$$

Me tango \frac{1}{2} mai i ngā taha e rua o te whārite.

$$x=18 x=-19$$