Whakaoti i te 8x^2-313x-615=0

$$8 x ^ { 2 } - 313 x - 615 = 0$$

Whakaoti mō x

$$x = -\frac{15}{8} = -1\frac{7}{8} = -1.875$$

$$x=41$$

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 8x^{2}+ax+bx-615. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-313 ab=8\left(-615\right)=-4920$$

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 -4920.

$$1,-4920 2,-2460 3,-1640 4,-1230 5,-984 6,-820 8,-615 10,-492 12,-410 15,-328 20,-246 24,-205 30,-164 40,-123 41,-120 60,-82$$

Tātaihia te tapeke mō ia takirua.

$$1-4920=-4919 2-2460=-2458 3-1640=-1637 4-1230=-1226 5-984=-979 6-820=-814 8-615=-607 10-492=-482 12-410=-398 15-328=-313 20-246=-226 24-205=-181 30-164=-134 40-123=-83 41-120=-79 60-82=-22$$

Ko te otinga te takirua ka hoatu i te tapeke -313.

$$a=-328 b=15$$

Tuhia anō te 8x^{2}-313x-615 hei \left(8x^{2}-328x\right)+\left(15x-615\right).

$$\left(8x^{2}-328x\right)+\left(15x-615\right)$$

Tauwehea te 8x i te tuatahi me te 15 i te rōpū tuarua.

$$8x\left(x-41\right)+15\left(x-41\right)$$

Whakatauwehea atu te kīanga pātahi x-41 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-41\right)\left(8x+15\right)$$

Hei kimi otinga whārite, me whakaoti te x-41=0 me te 8x+15=0.

$$x=41 x=-\frac{15}{8}$$

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.

$$8x^{2}-313x-615=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 8 mō a, -313 mō b, me -615 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-313\right)±\sqrt{\left(-313\right)^{2}-4\times 8\left(-615\right)}}{2\times 8}$$

Pūrua -313.

$$x=\frac{-\left(-313\right)±\sqrt{97969-4\times 8\left(-615\right)}}{2\times 8}$$

Whakareatia -4 ki te 8.

$$x=\frac{-\left(-313\right)±\sqrt{97969-32\left(-615\right)}}{2\times 8}$$

Whakareatia -32 ki te -615.

$$x=\frac{-\left(-313\right)±\sqrt{97969+19680}}{2\times 8}$$

Tāpiri 97969 ki te 19680.

$$x=\frac{-\left(-313\right)±\sqrt{117649}}{2\times 8}$$

Tuhia te pūtakerua o te 117649.

$$x=\frac{-\left(-313\right)±343}{2\times 8}$$

Ko te tauaro o -313 ko 313.

$$x=\frac{313±343}{2\times 8}$$

Whakareatia 2 ki te 8.

$$x=\frac{313±343}{16}$$

Nā, me whakaoti te whārite x=\frac{313±343}{16} ina he tāpiri te ±. Tāpiri 313 ki te 343.

$$x=\frac{656}{16}$$

Whakawehe 656 ki te 16.

$$x=41$$

Nā, me whakaoti te whārite x=\frac{313±343}{16} ina he tango te ±. Tango 343 mai i 313.

$$x=-\frac{30}{16}$$

Whakahekea te hautanga \frac{-30}{16} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x=-\frac{15}{8}$$

Kua oti te whārite te whakatau.

$$x=41 x=-\frac{15}{8}$$

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.

$$8x^{2}-313x-615=0$$

Me tāpiri 615 ki ngā taha e rua o te whārite.

$$8x^{2}-313x-615-\left(-615\right)=-\left(-615\right)$$

Mā te tango i te -615 i a ia ake anō ka toe ko te 0.

$$8x^{2}-313x=-\left(-615\right)$$

Tango -615 mai i 0.

$$8x^{2}-313x=615$$

Whakawehea ngā taha e rua ki te 8.

$$\frac{8x^{2}-313x}{8}=\frac{615}{8}$$

Mā te whakawehe ki te 8 ka wetekia te whakareanga ki te 8.

$$x^{2}-\frac{313}{8}x=\frac{615}{8}$$

Whakawehea te -\frac{313}{8}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{313}{16}. Nā, tāpiria te pūrua o te -\frac{313}{16} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{313}{8}x+\left(-\frac{313}{16}\right)^{2}=\frac{615}{8}+\left(-\frac{313}{16}\right)^{2}$$

Pūruatia -\frac{313}{16} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{313}{8}x+\frac{97969}{256}=\frac{615}{8}+\frac{97969}{256}$$

Tāpiri \frac{615}{8} ki te \frac{97969}{256} 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.

$$x^{2}-\frac{313}{8}x+\frac{97969}{256}=\frac{117649}{256}$$

Tauwehea x^{2}-\frac{313}{8}x+\frac{97969}{256}. 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}.

$$\left(x-\frac{313}{16}\right)^{2}=\frac{117649}{256}$$

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

$$\sqrt{\left(x-\frac{313}{16}\right)^{2}}=\sqrt{\frac{117649}{256}}$$

Whakarūnātia.

$$x-\frac{313}{16}=\frac{343}{16} x-\frac{313}{16}=-\frac{343}{16}$$

Me tāpiri \frac{313}{16} ki ngā taha e rua o te whārite.

$$x=41 x=-\frac{15}{8}$$