Whakaoti i te (7m-1)^2-100=0

$$( 7 m - 1 ) ^ { 2 } - 100 = 0$$

Whakaoti mō m

$$m = \frac{11}{7} = 1\frac{4}{7} \approx 1.571428571$$

$$m = -\frac{9}{7} = -1\frac{2}{7} \approx -1.285714286$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(7m-1\right)^{2}.

$$49m^{2}-14m+1-100=0$$

Tangohia te 100 i te 1, ka -99.

$$49m^{2}-14m-99=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 49m^{2}+am+bm-99. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-14 ab=49\left(-99\right)=-4851$$

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 -4851.

$$1,-4851 3,-1617 7,-693 9,-539 11,-441 21,-231 33,-147 49,-99 63,-77$$

Tātaihia te tapeke mō ia takirua.

$$1-4851=-4850 3-1617=-1614 7-693=-686 9-539=-530 11-441=-430 21-231=-210 33-147=-114 49-99=-50 63-77=-14$$

Ko te otinga te takirua ka hoatu i te tapeke -14.

$$a=-77 b=63$$

Tuhia anō te 49m^{2}-14m-99 hei \left(49m^{2}-77m\right)+\left(63m-99\right).

$$\left(49m^{2}-77m\right)+\left(63m-99\right)$$

Tauwehea te 7m i te tuatahi me te 9 i te rōpū tuarua.

$$7m\left(7m-11\right)+9\left(7m-11\right)$$

Whakatauwehea atu te kīanga pātahi 7m-11 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(7m-11\right)\left(7m+9\right)$$

Hei kimi otinga whārite, me whakaoti te 7m-11=0 me te 7m+9=0.

$$m=\frac{11}{7} m=-\frac{9}{7}$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(7m-1\right)^{2}.

$$49m^{2}-14m+1-100=0$$

Tangohia te 100 i te 1, ka -99.

$$49m^{2}-14m-99=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 49 mō a, -14 mō b, me -99 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$m=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\times 49\left(-99\right)}}{2\times 49}$$

Pūrua -14.

$$m=\frac{-\left(-14\right)±\sqrt{196-4\times 49\left(-99\right)}}{2\times 49}$$

Whakareatia -4 ki te 49.

$$m=\frac{-\left(-14\right)±\sqrt{196-196\left(-99\right)}}{2\times 49}$$

Whakareatia -196 ki te -99.

$$m=\frac{-\left(-14\right)±\sqrt{196+19404}}{2\times 49}$$

Tāpiri 196 ki te 19404.

$$m=\frac{-\left(-14\right)±\sqrt{19600}}{2\times 49}$$

Tuhia te pūtakerua o te 19600.

$$m=\frac{-\left(-14\right)±140}{2\times 49}$$

Ko te tauaro o -14 ko 14.

$$m=\frac{14±140}{2\times 49}$$

Whakareatia 2 ki te 49.

$$m=\frac{14±140}{98}$$

Nā, me whakaoti te whārite m=\frac{14±140}{98} ina he tāpiri te ±. Tāpiri 14 ki te 140.

$$m=\frac{154}{98}$$

Whakahekea te hautanga \frac{154}{98} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 14.

$$m=\frac{11}{7}$$

Nā, me whakaoti te whārite m=\frac{14±140}{98} ina he tango te ±. Tango 140 mai i 14.

$$m=-\frac{126}{98}$$

Whakahekea te hautanga \frac{-126}{98} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 14.

$$m=-\frac{9}{7}$$

Kua oti te whārite te whakatau.

$$m=\frac{11}{7} m=-\frac{9}{7}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(7m-1\right)^{2}.

$$49m^{2}-14m+1-100=0$$

Tangohia te 100 i te 1, ka -99.

$$49m^{2}-14m-99=0$$

Me tāpiri te 99 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$49m^{2}-14m=99$$

Whakawehea ngā taha e rua ki te 49.

$$\frac{49m^{2}-14m}{49}=\frac{99}{49}$$

Mā te whakawehe ki te 49 ka wetekia te whakareanga ki te 49.

$$m^{2}+\left(-\frac{14}{49}\right)m=\frac{99}{49}$$

Whakahekea te hautanga \frac{-14}{49} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 7.

$$m^{2}-\frac{2}{7}m=\frac{99}{49}$$

Whakawehea te -\frac{2}{7}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{1}{7}. Nā, tāpiria te pūrua o te -\frac{1}{7} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$m^{2}-\frac{2}{7}m+\left(-\frac{1}{7}\right)^{2}=\frac{99}{49}+\left(-\frac{1}{7}\right)^{2}$$

Pūruatia -\frac{1}{7} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$m^{2}-\frac{2}{7}m+\frac{1}{49}=\frac{99+1}{49}$$

Tāpiri \frac{99}{49} ki te \frac{1}{49} 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.

$$m^{2}-\frac{2}{7}m+\frac{1}{49}=\frac{100}{49}$$

Tauwehea m^{2}-\frac{2}{7}m+\frac{1}{49}. 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}.

$$\left(m-\frac{1}{7}\right)^{2}=\frac{100}{49}$$

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

$$\sqrt{\left(m-\frac{1}{7}\right)^{2}}=\sqrt{\frac{100}{49}}$$

Whakarūnātia.

$$m-\frac{1}{7}=\frac{10}{7} m-\frac{1}{7}=-\frac{10}{7}$$

Me tāpiri \frac{1}{7} ki ngā taha e rua o te whārite.

$$m=\frac{11}{7} m=-\frac{9}{7}$$