Whakaoti i te x^2+31x-360=0

$$x ^ { 2 } + 31 x - 360 = 0$$

Whakaoti mō x

$$x=-40$$

$$x=9$$

Ngā Upane Whakamahi Tauwehe

Hei whakaoti i te whārite, whakatauwehea te x^{2}+31x-360 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.

$$a+b=31 ab=-360$$

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 -360.

$$-1,360 -2,180 -3,120 -4,90 -5,72 -6,60 -8,45 -9,40 -10,36 -12,30 -15,24 -18,20$$

Tātaihia te tapeke mō ia takirua.

$$-1+360=359 -2+180=178 -3+120=117 -4+90=86 -5+72=67 -6+60=54 -8+45=37 -9+40=31 -10+36=26 -12+30=18 -15+24=9 -18+20=2$$

Ko te otinga te takirua ka hoatu i te tapeke 31.

$$a=-9 b=40$$

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

$$\left(x-9\right)\left(x+40\right)$$

Hei kimi otinga whārite, me whakaoti te x-9=0 me te x+40=0.

$$x=9 x=-40$$

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-360. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=31 ab=1\left(-360\right)=-360$$

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 -360.

$$-1,360 -2,180 -3,120 -4,90 -5,72 -6,60 -8,45 -9,40 -10,36 -12,30 -15,24 -18,20$$

Tātaihia te tapeke mō ia takirua.

$$-1+360=359 -2+180=178 -3+120=117 -4+90=86 -5+72=67 -6+60=54 -8+45=37 -9+40=31 -10+36=26 -12+30=18 -15+24=9 -18+20=2$$

Ko te otinga te takirua ka hoatu i te tapeke 31.

$$a=-9 b=40$$

Tuhia anō te x^{2}+31x-360 hei \left(x^{2}-9x\right)+\left(40x-360\right).

$$\left(x^{2}-9x\right)+\left(40x-360\right)$$

Tauwehea te x i te tuatahi me te 40 i te rōpū tuarua.

$$x\left(x-9\right)+40\left(x-9\right)$$

Whakatauwehea atu te kīanga pātahi x-9 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-9\right)\left(x+40\right)$$

Hei kimi otinga whārite, me whakaoti te x-9=0 me te x+40=0.

$$x=9 x=-40$$

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.

$$x^{2}+31x-360=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 31 mō b, me -360 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-31±\sqrt{31^{2}-4\left(-360\right)}}{2}$$

Pūrua 31.

$$x=\frac{-31±\sqrt{961-4\left(-360\right)}}{2}$$

Whakareatia -4 ki te -360.

$$x=\frac{-31±\sqrt{961+1440}}{2}$$

Tāpiri 961 ki te 1440.

$$x=\frac{-31±\sqrt{2401}}{2}$$

Tuhia te pūtakerua o te 2401.

$$x=\frac{-31±49}{2}$$

Nā, me whakaoti te whārite x=\frac{-31±49}{2} ina he tāpiri te ±. Tāpiri -31 ki te 49.

$$x=\frac{18}{2}$$

Whakawehe 18 ki te 2.

$$x=9$$

Nā, me whakaoti te whārite x=\frac{-31±49}{2} ina he tango te ±. Tango 49 mai i -31.

$$x=-\frac{80}{2}$$

Whakawehe -80 ki te 2.

$$x=-40$$

Kua oti te whārite te whakatau.

$$x=9 x=-40$$

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.

$$x^{2}+31x-360=0$$

Me tāpiri 360 ki ngā taha e rua o te whārite.

$$x^{2}+31x-360-\left(-360\right)=-\left(-360\right)$$

Mā te tango i te -360 i a ia ake anō ka toe ko te 0.

$$x^{2}+31x=-\left(-360\right)$$

Tango -360 mai i 0.

$$x^{2}+31x=360$$

Whakawehea te 31, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{31}{2}. Nā, tāpiria te pūrua o te \frac{31}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+31x+\left(\frac{31}{2}\right)^{2}=360+\left(\frac{31}{2}\right)^{2}$$

Pūruatia \frac{31}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+31x+\frac{961}{4}=360+\frac{961}{4}$$

Tāpiri 360 ki te \frac{961}{4}.

$$x^{2}+31x+\frac{961}{4}=\frac{2401}{4}$$

Tauwehea x^{2}+31x+\frac{961}{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}.

$$\left(x+\frac{31}{2}\right)^{2}=\frac{2401}{4}$$

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

$$\sqrt{\left(x+\frac{31}{2}\right)^{2}}=\sqrt{\frac{2401}{4}}$$

Whakarūnātia.

$$x+\frac{31}{2}=\frac{49}{2} x+\frac{31}{2}=-\frac{49}{2}$$

Me tango \frac{31}{2} mai i ngā taha e rua o te whārite.

$$x=9 x=-40$$