Whakaoti i te 11x^2-98x-120=0

$$11 x ^ { 2 } - 98 x - 120 = 0$$

Whakaoti mō x

$$x = -\frac{12}{11} = -1\frac{1}{11} \approx -1.090909091$$

$$x=10$$

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 11x^{2}+ax+bx-120. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-98 ab=11\left(-120\right)=-1320$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōraro te a+b, he nui ake te uara pū o te tau tōraro i tō te tōrunga. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -1320.

$$1,-1320 2,-660 3,-440 4,-330 5,-264 6,-220 8,-165 10,-132 11,-120 12,-110 15,-88 20,-66 22,-60 24,-55 30,-44 33,-40$$

Tātaihia te tapeke mō ia takirua.

$$1-1320=-1319 2-660=-658 3-440=-437 4-330=-326 5-264=-259 6-220=-214 8-165=-157 10-132=-122 11-120=-109 12-110=-98 15-88=-73 20-66=-46 22-60=-38 24-55=-31 30-44=-14 33-40=-7$$

Ko te otinga te takirua ka hoatu i te tapeke -98.

$$a=-110 b=12$$

Tuhia anō te 11x^{2}-98x-120 hei \left(11x^{2}-110x\right)+\left(12x-120\right).

$$\left(11x^{2}-110x\right)+\left(12x-120\right)$$

Tauwehea te 11x i te tuatahi me te 12 i te rōpū tuarua.

$$11x\left(x-10\right)+12\left(x-10\right)$$

Whakatauwehea atu te kīanga pātahi x-10 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-10\right)\left(11x+12\right)$$

Hei kimi otinga whārite, me whakaoti te x-10=0 me te 11x+12=0.

$$x=10 x=-\frac{12}{11}$$

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.

$$11x^{2}-98x-120=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 11 mō a, -98 mō b, me -120 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-98\right)±\sqrt{\left(-98\right)^{2}-4\times 11\left(-120\right)}}{2\times 11}$$

Pūrua -98.

$$x=\frac{-\left(-98\right)±\sqrt{9604-4\times 11\left(-120\right)}}{2\times 11}$$

Whakareatia -4 ki te 11.

$$x=\frac{-\left(-98\right)±\sqrt{9604-44\left(-120\right)}}{2\times 11}$$

Whakareatia -44 ki te -120.

$$x=\frac{-\left(-98\right)±\sqrt{9604+5280}}{2\times 11}$$

Tāpiri 9604 ki te 5280.

$$x=\frac{-\left(-98\right)±\sqrt{14884}}{2\times 11}$$

Tuhia te pūtakerua o te 14884.

$$x=\frac{-\left(-98\right)±122}{2\times 11}$$

Ko te tauaro o -98 ko 98.

$$x=\frac{98±122}{2\times 11}$$

Whakareatia 2 ki te 11.

$$x=\frac{98±122}{22}$$

Nā, me whakaoti te whārite x=\frac{98±122}{22} ina he tāpiri te ±. Tāpiri 98 ki te 122.

$$x=\frac{220}{22}$$

Whakawehe 220 ki te 22.

$$x=10$$

Nā, me whakaoti te whārite x=\frac{98±122}{22} ina he tango te ±. Tango 122 mai i 98.

$$x=-\frac{24}{22}$$

Whakahekea te hautanga \frac{-24}{22} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x=-\frac{12}{11}$$

Kua oti te whārite te whakatau.

$$x=10 x=-\frac{12}{11}$$

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.

$$11x^{2}-98x-120=0$$

Me tāpiri 120 ki ngā taha e rua o te whārite.

$$11x^{2}-98x-120-\left(-120\right)=-\left(-120\right)$$

Mā te tango i te -120 i a ia ake anō ka toe ko te 0.

$$11x^{2}-98x=-\left(-120\right)$$

Tango -120 mai i 0.

$$11x^{2}-98x=120$$

Whakawehea ngā taha e rua ki te 11.

$$\frac{11x^{2}-98x}{11}=\frac{120}{11}$$

Mā te whakawehe ki te 11 ka wetekia te whakareanga ki te 11.

$$x^{2}-\frac{98}{11}x=\frac{120}{11}$$

Whakawehea te -\frac{98}{11}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{49}{11}. Nā, tāpiria te pūrua o te -\frac{49}{11} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{98}{11}x+\left(-\frac{49}{11}\right)^{2}=\frac{120}{11}+\left(-\frac{49}{11}\right)^{2}$$

Pūruatia -\frac{49}{11} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{98}{11}x+\frac{2401}{121}=\frac{120}{11}+\frac{2401}{121}$$

Tāpiri \frac{120}{11} ki te \frac{2401}{121} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x^{2}-\frac{98}{11}x+\frac{2401}{121}=\frac{3721}{121}$$

Tauwehea x^{2}-\frac{98}{11}x+\frac{2401}{121}. 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}.

$$\left(x-\frac{49}{11}\right)^{2}=\frac{3721}{121}$$

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

$$\sqrt{\left(x-\frac{49}{11}\right)^{2}}=\sqrt{\frac{3721}{121}}$$

Whakarūnātia.

$$x-\frac{49}{11}=\frac{61}{11} x-\frac{49}{11}=-\frac{61}{11}$$

Me tāpiri \frac{49}{11} ki ngā taha e rua o te whārite.

$$x=10 x=-\frac{12}{11}$$