Whakaoti i te 1034=x^2+30x-110

$$1034 = x ^ { 2 } + 30 x - 110$$

Whakaoti mō x

$$x=-52$$

$$x=22$$

Ngā Upane Whakamahi Tauwehe

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$x^{2}+30x-110=1034$$

Tangohia te 1034 mai i ngā taha e rua.

$$x^{2}+30x-110-1034=0$$

Tangohia te 1034 i te -110, ka -1144.

$$x^{2}+30x-1144=0$$

Hei whakaoti i te whārite, whakatauwehea te x^{2}+30x-1144 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.

$$a+b=30 ab=-1144$$

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 -1144.

$$-1,1144 -2,572 -4,286 -8,143 -11,104 -13,88 -22,52 -26,44$$

Tātaihia te tapeke mō ia takirua.

$$-1+1144=1143 -2+572=570 -4+286=282 -8+143=135 -11+104=93 -13+88=75 -22+52=30 -26+44=18$$

Ko te otinga te takirua ka hoatu i te tapeke 30.

$$a=-22 b=52$$

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

$$\left(x-22\right)\left(x+52\right)$$

Hei kimi otinga whārite, me whakaoti te x-22=0 me te x+52=0.

$$x=22 x=-52$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$x^{2}+30x-110=1034$$

Tangohia te 1034 mai i ngā taha e rua.

$$x^{2}+30x-110-1034=0$$

Tangohia te 1034 i te -110, ka -1144.

$$x^{2}+30x-1144=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx-1144. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=30 ab=1\left(-1144\right)=-1144$$

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 -1144.

$$-1,1144 -2,572 -4,286 -8,143 -11,104 -13,88 -22,52 -26,44$$

Tātaihia te tapeke mō ia takirua.

$$-1+1144=1143 -2+572=570 -4+286=282 -8+143=135 -11+104=93 -13+88=75 -22+52=30 -26+44=18$$

Ko te otinga te takirua ka hoatu i te tapeke 30.

$$a=-22 b=52$$

Tuhia anō te x^{2}+30x-1144 hei \left(x^{2}-22x\right)+\left(52x-1144\right).

$$\left(x^{2}-22x\right)+\left(52x-1144\right)$$

Tauwehea te x i te tuatahi me te 52 i te rōpū tuarua.

$$x\left(x-22\right)+52\left(x-22\right)$$

Whakatauwehea atu te kīanga pātahi x-22 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-22\right)\left(x+52\right)$$

Hei kimi otinga whārite, me whakaoti te x-22=0 me te x+52=0.

$$x=22 x=-52$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$x^{2}+30x-110=1034$$

Tangohia te 1034 mai i ngā taha e rua.

$$x^{2}+30x-110-1034=0$$

Tangohia te 1034 i te -110, ka -1144.

$$x^{2}+30x-1144=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 30 mō b, me -1144 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-30±\sqrt{30^{2}-4\left(-1144\right)}}{2}$$

Pūrua 30.

$$x=\frac{-30±\sqrt{900-4\left(-1144\right)}}{2}$$

Whakareatia -4 ki te -1144.

$$x=\frac{-30±\sqrt{900+4576}}{2}$$

Tāpiri 900 ki te 4576.

$$x=\frac{-30±\sqrt{5476}}{2}$$

Tuhia te pūtakerua o te 5476.

$$x=\frac{-30±74}{2}$$

Nā, me whakaoti te whārite x=\frac{-30±74}{2} ina he tāpiri te ±. Tāpiri -30 ki te 74.

$$x=\frac{44}{2}$$

Whakawehe 44 ki te 2.

$$x=22$$

Nā, me whakaoti te whārite x=\frac{-30±74}{2} ina he tango te ±. Tango 74 mai i -30.

$$x=-\frac{104}{2}$$

Whakawehe -104 ki te 2.

$$x=-52$$

Kua oti te whārite te whakatau.

$$x=22 x=-52$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$x^{2}+30x-110=1034$$

Me tāpiri te 110 ki ngā taha e rua.

$$x^{2}+30x=1034+110$$

Tāpirihia te 1034 ki te 110, ka 1144.

$$x^{2}+30x=1144$$

Whakawehea te 30, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 15. Nā, tāpiria te pūrua o te 15 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+30x+15^{2}=1144+15^{2}$$

Pūrua 15.

$$x^{2}+30x+225=1144+225$$

Tāpiri 1144 ki te 225.

$$x^{2}+30x+225=1369$$

Tauwehea x^{2}+30x+225. 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}.

$$\left(x+15\right)^{2}=1369$$

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

$$\sqrt{\left(x+15\right)^{2}}=\sqrt{1369}$$

Whakarūnātia.

$$x+15=37 x+15=-37$$

Me tango 15 mai i ngā taha e rua o te whārite.

$$x=22 x=-52$$