Whakaoti i te 2b^2-105=-11b

$$2 b ^ { 2 } - 105 = - 11 b$$

Whakaoti mō b

$$b = -\frac{21}{2} = -10\frac{1}{2} = -10.5$$

$$b=5$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Me tāpiri te 11b ki ngā taha e rua.

$$2b^{2}-105+11b=0$$

Hurinahatia te pūrau ki te āhua tānga ngahuru. Whakaraupapahia ngā kīanga tau mai i te pū teitei rawa ki te mea iti rawa.

$$2b^{2}+11b-105=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 2b^{2}+ab+bb-105. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=11 ab=2\left(-105\right)=-210$$

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 -210.

$$-1,210 -2,105 -3,70 -5,42 -6,35 -7,30 -10,21 -14,15$$

Tātaihia te tapeke mō ia takirua.

$$-1+210=209 -2+105=103 -3+70=67 -5+42=37 -6+35=29 -7+30=23 -10+21=11 -14+15=1$$

Ko te otinga te takirua ka hoatu i te tapeke 11.

$$a=-10 b=21$$

Tuhia anō te 2b^{2}+11b-105 hei \left(2b^{2}-10b\right)+\left(21b-105\right).

$$\left(2b^{2}-10b\right)+\left(21b-105\right)$$

Tauwehea te 2b i te tuatahi me te 21 i te rōpū tuarua.

$$2b\left(b-5\right)+21\left(b-5\right)$$

Whakatauwehea atu te kīanga pātahi b-5 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(b-5\right)\left(2b+21\right)$$

Hei kimi otinga whārite, me whakaoti te b-5=0 me te 2b+21=0.

$$b=5 b=-\frac{21}{2}$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Me tāpiri te 11b ki ngā taha e rua.

$$2b^{2}-105+11b=0$$

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.

$$2b^{2}+11b-105=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 2 mō a, 11 mō b, me -105 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$b=\frac{-11±\sqrt{11^{2}-4\times 2\left(-105\right)}}{2\times 2}$$

Pūrua 11.

$$b=\frac{-11±\sqrt{121-4\times 2\left(-105\right)}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$b=\frac{-11±\sqrt{121-8\left(-105\right)}}{2\times 2}$$

Whakareatia -8 ki te -105.

$$b=\frac{-11±\sqrt{121+840}}{2\times 2}$$

Tāpiri 121 ki te 840.

$$b=\frac{-11±\sqrt{961}}{2\times 2}$$

Tuhia te pūtakerua o te 961.

$$b=\frac{-11±31}{2\times 2}$$

Whakareatia 2 ki te 2.

$$b=\frac{-11±31}{4}$$

Nā, me whakaoti te whārite b=\frac{-11±31}{4} ina he tāpiri te ±. Tāpiri -11 ki te 31.

$$b=\frac{20}{4}$$

Whakawehe 20 ki te 4.

$$b=5$$

Nā, me whakaoti te whārite b=\frac{-11±31}{4} ina he tango te ±. Tango 31 mai i -11.

$$b=-\frac{42}{4}$$

Whakahekea te hautanga \frac{-42}{4} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$b=-\frac{21}{2}$$

Kua oti te whārite te whakatau.

$$b=5 b=-\frac{21}{2}$$

Ngā Upane mō te Whakaoti i te Pūrua

Me tāpiri te 11b ki ngā taha e rua.

$$2b^{2}-105+11b=0$$

Me tāpiri te 105 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$2b^{2}+11b=105$$

Whakawehea ngā taha e rua ki te 2.

$$\frac{2b^{2}+11b}{2}=\frac{105}{2}$$

Mā te whakawehe ki te 2 ka wetekia te whakareanga ki te 2.

$$b^{2}+\frac{11}{2}b=\frac{105}{2}$$

Whakawehea te \frac{11}{2}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{11}{4}. Nā, tāpiria te pūrua o te \frac{11}{4} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$b^{2}+\frac{11}{2}b+\left(\frac{11}{4}\right)^{2}=\frac{105}{2}+\left(\frac{11}{4}\right)^{2}$$

Pūruatia \frac{11}{4} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$b^{2}+\frac{11}{2}b+\frac{121}{16}=\frac{105}{2}+\frac{121}{16}$$

Tāpiri \frac{105}{2} ki te \frac{121}{16} 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.

$$b^{2}+\frac{11}{2}b+\frac{121}{16}=\frac{961}{16}$$

Tauwehea b^{2}+\frac{11}{2}b+\frac{121}{16}. 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}.

$$\left(b+\frac{11}{4}\right)^{2}=\frac{961}{16}$$

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

$$\sqrt{\left(b+\frac{11}{4}\right)^{2}}=\sqrt{\frac{961}{16}}$$

Whakarūnātia.

$$b+\frac{11}{4}=\frac{31}{4} b+\frac{11}{4}=-\frac{31}{4}$$

Me tango \frac{11}{4} mai i ngā taha e rua o te whārite.

$$b=5 b=-\frac{21}{2}$$