Whakaoti i te x^2-x^-9900=0

$${ x }^{ 2 } - { x }^{ } -9900=0$$

Whakaoti mō x

$$x=-99$$

$$x=100$$

Ngā Upane Whakamahi Tauwehe

Tātaihia te x mā te pū o 1, kia riro ko x.

$$x^{2}-x-9900=0$$

Hei whakaoti i te whārite, whakatauwehea te x^{2}-x-9900 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.

$$a+b=-1 ab=-9900$$

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 -9900.

$$1,-9900 2,-4950 3,-3300 4,-2475 5,-1980 6,-1650 9,-1100 10,-990 11,-900 12,-825 15,-660 18,-550 20,-495 22,-450 25,-396 30,-330 33,-300 36,-275 44,-225 45,-220 50,-198 55,-180 60,-165 66,-150 75,-132 90,-110 99,-100$$

Tātaihia te tapeke mō ia takirua.

$$1-9900=-9899 2-4950=-4948 3-3300=-3297 4-2475=-2471 5-1980=-1975 6-1650=-1644 9-1100=-1091 10-990=-980 11-900=-889 12-825=-813 15-660=-645 18-550=-532 20-495=-475 22-450=-428 25-396=-371 30-330=-300 33-300=-267 36-275=-239 44-225=-181 45-220=-175 50-198=-148 55-180=-125 60-165=-105 66-150=-84 75-132=-57 90-110=-20 99-100=-1$$

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

$$a=-100 b=99$$

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

$$\left(x-100\right)\left(x+99\right)$$

Hei kimi otinga whārite, me whakaoti te x-100=0 me te x+99=0.

$$x=100 x=-99$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tātaihia te x mā te pū o 1, kia riro ko x.

$$x^{2}-x-9900=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx-9900. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-1 ab=1\left(-9900\right)=-9900$$

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 -9900.

$$1,-9900 2,-4950 3,-3300 4,-2475 5,-1980 6,-1650 9,-1100 10,-990 11,-900 12,-825 15,-660 18,-550 20,-495 22,-450 25,-396 30,-330 33,-300 36,-275 44,-225 45,-220 50,-198 55,-180 60,-165 66,-150 75,-132 90,-110 99,-100$$

Tātaihia te tapeke mō ia takirua.

$$1-9900=-9899 2-4950=-4948 3-3300=-3297 4-2475=-2471 5-1980=-1975 6-1650=-1644 9-1100=-1091 10-990=-980 11-900=-889 12-825=-813 15-660=-645 18-550=-532 20-495=-475 22-450=-428 25-396=-371 30-330=-300 33-300=-267 36-275=-239 44-225=-181 45-220=-175 50-198=-148 55-180=-125 60-165=-105 66-150=-84 75-132=-57 90-110=-20 99-100=-1$$

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

$$a=-100 b=99$$

Tuhia anō te x^{2}-x-9900 hei \left(x^{2}-100x\right)+\left(99x-9900\right).

$$\left(x^{2}-100x\right)+\left(99x-9900\right)$$

Tauwehea te x i te tuatahi me te 99 i te rōpū tuarua.

$$x\left(x-100\right)+99\left(x-100\right)$$

Whakatauwehea atu te kīanga pātahi x-100 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-100\right)\left(x+99\right)$$

Hei kimi otinga whārite, me whakaoti te x-100=0 me te x+99=0.

$$x=100 x=-99$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tātaihia te x mā te pū o 1, kia riro ko x.

$$x^{2}-x-9900=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -1 mō b, me -9900 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\left(-9900\right)}}{2}$$

Whakareatia -4 ki te -9900.

$$x=\frac{-\left(-1\right)±\sqrt{1+39600}}{2}$$

Tāpiri 1 ki te 39600.

$$x=\frac{-\left(-1\right)±\sqrt{39601}}{2}$$

Tuhia te pūtakerua o te 39601.

$$x=\frac{-\left(-1\right)±199}{2}$$

Ko te tauaro o -1 ko 1.

$$x=\frac{1±199}{2}$$

Nā, me whakaoti te whārite x=\frac{1±199}{2} ina he tāpiri te ±. Tāpiri 1 ki te 199.

$$x=\frac{200}{2}$$

Whakawehe 200 ki te 2.

$$x=100$$

Nā, me whakaoti te whārite x=\frac{1±199}{2} ina he tango te ±. Tango 199 mai i 1.

$$x=-\frac{198}{2}$$

Whakawehe -198 ki te 2.

$$x=-99$$

Kua oti te whārite te whakatau.

$$x=100 x=-99$$

Ngā Upane mō te Whakaoti i te Pūrua

Tātaihia te x mā te pū o 1, kia riro ko x.

$$x^{2}-x-9900=0$$

Me tāpiri te 9900 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$x^{2}-x=9900$$

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.

$$x^{2}-x+\left(-\frac{1}{2}\right)^{2}=9900+\left(-\frac{1}{2}\right)^{2}$$

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

$$x^{2}-x+\frac{1}{4}=9900+\frac{1}{4}$$

Tāpiri 9900 ki te \frac{1}{4}.

$$x^{2}-x+\frac{1}{4}=\frac{39601}{4}$$

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}.

$$\left(x-\frac{1}{2}\right)^{2}=\frac{39601}{4}$$

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

$$\sqrt{\left(x-\frac{1}{2}\right)^{2}}=\sqrt{\frac{39601}{4}}$$

Whakarūnātia.

$$x-\frac{1}{2}=\frac{199}{2} x-\frac{1}{2}=-\frac{199}{2}$$

Me tāpiri \frac{1}{2} ki ngā taha e rua o te whārite.

$$x=100 x=-99$$